Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression \({a}^{m},\) the exponent m tells us how many times we use the base a as a factor.

Let’s review the vocabulary for expressions with exponents.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property.

		\(\\)
What does this mean?	\(\\)
		\(\\)

Notice that 5 is the sum of the exponents, 2 and 3. We see \({x}^{2}\cdot {x}^{3}\) is \({x}^{2+3}\) or \({x}^{5}.\)

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Example

Try it.

Simplify each expression: ⓐ \({y}^{5}\cdot {y}^{6}\) ⓑ \({2}^{x}\cdot {2}^{3x}\) ⓒ \(2{a}^{7}\cdot 3a.\) ⓓ \({d}^{4}⋅{d}^{5}⋅{d}^{2}\)

Solution

ⓐ

Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify.

ⓑ

Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify.

ⓒ

Rewrite, \(a={a}^{1}.\)
Use the Commutative Property and use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
Simplify.

ⓓ

Add the exponents, since bases are the same.
Simplify.

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

Consider	\(\frac{{x}^{5}}{{x}^{2}}\)	and	\(\frac{{x}^{2}}{{x}^{3}}\)
What do they mean?	\(\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\)		\(\frac{x\cdot x}{x\cdot x\cdot x}\)
Use the Equivalent Fractions Property.	\(\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\)		\(\frac{x\cdot x\cdot 1}{x\cdot x\cdot x}\)
Simplify.	\({x}^{3}\)		\(\frac{1}{x}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Let’s consider \(\frac{{x}^{2}}{{x}^{5}}.\) We subtract the exponent in the denominator from the exponent in the numerator. We see \(\frac{{x}^{2}}{{x}^{5}}\) is \({x}^{2-5}\) or \({x}^{-3}.\)

We can also simplify \(\frac{{x}^{2}}{{x}^{5}}\) by dividing out common factors:

This implies that \({x}^{-3}=\frac{1}{{x}^{3}}\) and it leads us to the definition of a negative exponent. If n is an integer and \(a\ne 0,\) then \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)

Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

	\(\ \frac{1}{{a}^{\text{-}n}}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{\frac{1}{{a}^{n}}}\)
Simplify the complex fraction.	\(\ 1\cdot \frac{{a}^{n}}{1}\)
Multiply.	\(\ {a}^{n}\)

This implies \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\) and is another form of the definition of Properties of Negative Exponents.

Example

Try it.

Simplify each expression: ⓐ \({x}^{-5}\) ⓑ \({10}^{-3}\) ⓒ \(\frac{1}{{y}^{-4}}\) ⓓ \(\frac{1}{{3}^{-2}}.\)

Solution

ⓐ

	\(\ {x}^{-5}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{{x}^{5}}\)

ⓑ

	\(\ {10}^{-3}\)
Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{{10}^{3}}\)
Simplify.	\(\ \frac{1}{1000}\)

ⓒ

	\(\ \frac{1}{{y}^{-4}}\)
Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}.\)	\(\ {y}^{4}\)

ⓓ

	\(\ \frac{1}{{3}^{-2}}\)
Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}.\)	\(\ {3}^{2}\)
Simplify.	\(\ 9\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means \(4\ \times \ 1,000\) and 0.004 means \(4\ \times \ \frac{1}{1,000}.\)

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

4,000	\(4\ \times \ 1,000\)	\(4\ \times \ {10}^{3}\)
0.004	\(4\ \times \ \frac{1}{1,000}\)	\(4\ \times \ \frac{1}{{10}^{3}}\)	\(4\ \times \ {10}^{-3}\)

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than ten, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

It is customary in scientific notation to use as the \(\ \times \\) multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

\[\begin{array}{llllll}9.12\ \times \ {10}^{4} & & & & & 9.12\ \times \ {10}^{-4} \\ 9.12\ \times \ 10,000 & & & & & 9.12\ \times \ 0.0001 \\ 91,200 & & & & & 0.000912\end{array}\]

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

• Exponential Notation
  
  This is read a to the \({m}^{th}\) power.
  In the expression \({a}^{m}\), the exponent m tells us how many times we use the base a as a factor.
• Product Property for Exponents
  If a is a real number and m and n are integers, then
  \[{a}^{m}\cdot {a}^{n}={a}^{m+n}\]
  To multiply with like bases, add the exponents.
• Quotient Property for Exponents
  If \(a\) is a real number, \(a\ne 0,\) and m and n are integers, then
  \[\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\ \ m>n\ \text{and}\ \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\ n>m\]
• Zero Exponent
  • If a is a non-zero number, then \({a}^{0}=1.\)
  • If a is a non-zero number, then a to the power of zero equals 1.
  • Any non-zero number raised to the zero power is 1.
• Negative Exponent
  • If n is an integer and \(a\ne 0,\) then \({a}^{\text{-}n}=\frac{1}{{a}^{n}}\) or \(\frac{1}{{a}^{\text{-}n}}={a}^{n}.\)
• Quotient to a Negative Exponent Property
  If \(a,b\) are real numbers, \(a\ne 0,b\ne 0\) and \(n\) is an integer, then
  \[{(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\]
• Power Property for Exponents
  If \(a\) is a real number and \(m,n\) are integers, then
  \[{({a}^{m})}^{n}={a}^{m\cdot n}\]
  To raise a power to a power, multiply the exponents.
• Product to a Power Property for Exponents
  If a and b are real numbers and m is a whole number, then
  \[{(ab)}^{m}={a}^{m}{b}^{m}\]
  To raise a product to a power, raise each factor to that power.
• Quotient to a Power Property for Exponents
  If \(a\) and are real numbers, \(b\ne 0,\) and \(m\) is an integer, then
  \[{(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\]
  To raise a fraction to a power, raise the numerator and denominator to that power.
• Summary of Exponent Properties
  If a and b are real numbers, and m and n are integers, then
  
  

  Property	Description
  Product Property	\({a}^{m}\cdot {a}^{n}={a}^{m+n}\)
  Power Property	\({({a}^{m})}^{n}={a}^{m\cdot n}\)
  Product to a Power	\({(ab)}^{n}={a}^{n}{b}^{n}\)
  Quotient Property	\(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},a\ne 0\)
  Zero Exponent Property	\({a}^{0}=1,a\ne 0\)
  Quotient to a Power Property:	\({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\)
  Properties of Negative Exponents	\({a}^{\text{-}n}=\frac{1}{{a}^{n}}\) and \(\frac{1}{{a}^{\text{-}n}}={a}^{n}\)
  Quotient to a Negative Exponent	\({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}\)

• Scientific Notation
  A number is expressed in scientific notation when it is of the form
  \[a\ \times \ {10}^{n}\ \text{where}\ 1\le a<10\ \text{and}\ n\ \text{is an integer.}\]
• How to convert a decimal to scientific notation.
  1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
  2. Count the number of decimal places, \(n,\) that the decimal point was moved.
  3. Write the number as a product with a power of 10. If the original number is.
     • greater than 1, the power of 10 will be \({10}^{n}.\)
     • between 0 and 1, the power of 10 will be \({10}^{\text{-}n}.\)
  4. Check.
• How to convert scientific notation to decimal form.
  1. Determine the exponent, \(n,\) on the factor 10.
  2. Move the decimal \(n\) places, adding zeros if needed.
     • If the exponent is positive, move the decimal point \(n\) places to the right.
     • If the exponent is negative, move the decimal point \(|n|\) places to the left.
  3. Check.

Properties of Exponents and Scientific Notation

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

Try it.

ⓐ \({d}^{3}\cdot {d}^{6}\) ⓑ \({4}^{5x}\cdot {4}^{9x}\) ⓒ \(2y\cdot 4{y}^{3}\) ⓓ \(w\cdot {w}^{2}\cdot {w}^{3}\)

Solution

ⓐ \({d}^{9}\) ⓑ \({4}^{14x}\) ⓒ \(8{y}^{4}\) ⓓ \({w}^{6}\)

Try it.

ⓐ \({x}^{4}\cdot {x}^{2}\) ⓑ \({8}^{9x}\cdot {8}^{3}\) ⓒ \(3{z}^{25}\cdot 5{z}^{8}\) ⓓ \(y\cdot {y}^{3}\cdot {y}^{5}\)

Try it.

ⓐ \({n}^{19}\cdot {n}^{12}\) ⓑ \({3}^{x}\cdot {3}^{6}\) ⓒ \(7{w}^{5}\cdot 8{w}^{}\) ⓓ \({a}^{4}\cdot {a}^{3}\cdot {a}^{9}\)

Solution

ⓐ \({n}^{31}\) ⓑ \({3}^{x+6}\) ⓒ \(56{w}^{6}\)
ⓓ \({a}^{16}\)

Try it.

ⓐ \({q}^{27}\cdot {q}^{15}\) ⓑ \({5}^{x}\cdot {5}^{4x}\) ⓒ \(9{u}^{41}\cdot 7{u}^{53}\)
ⓓ \({c}^{5}\cdot {c}^{11}\cdot {c}^{2}\)

Try it.

\({m}^{x}\cdot {m}^{3}\)

Solution

\({m}^{x+3}\)

Try it.

\({n}^{y}\cdot {n}^{2}\)

Try it.

\({y}^{a}\cdot {y}^{b}\)

Solution

\({y}^{a+b}\)

Try it.

\({x}^{p}\cdot {x}^{q}\)

Try it.

ⓐ \(\frac{{x}^{18}}{{x}^{3}}\) ⓑ \(\frac{{5}^{12}}{{5}^{3}}\) ⓒ \(\frac{{q}^{18}}{{q}^{36}}\) ⓓ \(\frac{{10}^{2}}{{10}^{3}}\)

Solution

ⓐ \({x}^{15}\) ⓑ \({5}^{9}\) ⓒ \(\frac{1}{{q}^{18}}\) ⓓ \(\frac{1}{10}\)

Try it.

ⓐ \(\frac{{y}^{20}}{{y}^{10}}\) ⓑ \(\frac{{7}^{16}}{{7}^{2}}\) ⓒ \(\frac{{t}^{10}}{{t}^{40}}\) ⓓ \(\frac{{8}^{3}}{{8}^{5}}\)

Try it.

ⓐ \(\frac{{p}^{21}}{{p}^{7}}\) ⓑ \(\frac{{4}^{16}}{{4}^{4}}\) ⓒ \(\frac{b}{{b}^{9}}\) ⓓ \(\frac{4}{{4}^{6}}\)

Solution

ⓐ \({p}^{14}\) ⓑ \({4}^{12}\) ⓒ \(\frac{1}{{b}^{8}}\) ⓓ \(\frac{1}{{4}^{5}}\)

Try it.

ⓐ \(\frac{{u}^{24}}{{u}^{3}}\) ⓑ \(\frac{{9}^{15}}{{9}^{5}}\) ⓒ \(\frac{x}{{x}^{7}}\) ⓓ \(\frac{10}{{10}^{3}}\)

Try it.

ⓐ \({20}^{0}\) ⓑ \({b}^{0}\)

Solution

ⓐ 1 ⓑ 1

Try it.

ⓐ \({13}^{0}\) ⓑ \({k}^{0}\)

Try it.

ⓐ \(\text{-}{27}^{0}\) ⓑ \(\text{-}({27}^{0})\)

Solution

ⓐ \(-1\) ⓑ \(-1\)

Try it.

ⓐ \(\text{-}{15}^{0}\) ⓑ \(\text{-}({15}^{0})\)

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

Try it.

ⓐ \({a}^{-2}\) ⓑ \({10}^{-3}\) ⓒ \(\frac{1}{{c}^{-5}}\) ⓓ \(\frac{1}{{3}^{-2}}\)

Solution

ⓐ \(\frac{1}{{a}^{2}}\) ⓑ \(\frac{1}{1000}\) ⓒ \({c}^{5}\) ⓓ \(9\)

Try it.

ⓐ \({b}^{-4}\) ⓑ \({10}^{-2}\) ⓒ \(\frac{1}{{b}^{-3}}\) ⓓ \(\frac{1}{{5}^{-2}}\)

Try it.

ⓐ \({r}^{-3}\) ⓑ \({10}^{-5}\) ⓒ \(\frac{1}{{q}^{-10}}\) ⓓ \(\frac{1}{{10}^{-3}}\)

Solution

ⓐ \(\frac{1}{{r}^{3}}\) ⓑ \(\frac{1}{100,000}\) ⓒ \({q}^{10}\)
ⓓ \(1,000\)

Try it.

ⓐ \({s}^{-8}\) ⓑ \({10}^{-2}\) ⓒ \(\frac{1}{{t}^{-9}}\) ⓓ \(\frac{1}{{10}^{-4}}\)

Try it.

ⓐ \({(\frac{5}{8})}^{-2}\) ⓑ \({(-\frac{b}{a})}^{-2}\)

Solution

ⓐ \(\frac{64}{25}\) ⓑ \(\frac{{a}^{2}}{{b}^{2}}\)

Try it.

ⓐ \({(\frac{3}{10})}^{-2}\) ⓑ \({(-\frac{2}{z})}^{-3}\)

Try it.

ⓐ \({(\frac{4}{9})}^{-3}\) ⓑ \({(-\frac{u}{v})}^{-5}\)

Solution

ⓐ \(\frac{729}{64}\) ⓑ \(-\frac{{v}^{5}}{{u}^{5}}\)

Try it.

ⓐ \({(\frac{7}{2})}^{-3}\) ⓑ \({(-\frac{3}{x})}^{-3}\)

Try it.

ⓐ \({(-5)}^{-2}\) ⓑ \(\text{-}{5}^{-2}\) ⓒ \({(-\frac{1}{5})}^{-2}\) ⓓ \(\text{-}{(\frac{1}{5})}^{-2}\)

Solution

ⓐ \(\frac{1}{25}\) ⓑ \(-\frac{1}{25}\) ⓒ \(25\) ⓓ \(-25\)

Try it.

ⓐ \(\text{-}{5}^{-3}\) ⓑ \({(-\frac{1}{5})}^{-3}\) ⓒ \(\text{-}{(\frac{1}{5})}^{-3}\) ⓓ \({(-5)}^{-3}\)

Try it.

ⓐ \(3\cdot {5}^{-1}\) ⓑ \({(3\cdot 5)}^{-1}\)

Solution

ⓐ \(\frac{3}{5}\) ⓑ \(\frac{1}{15}\)

Try it.

ⓐ \(3\cdot {4}^{-2}\) ⓑ \({(3\cdot 4)}^{-2}\)

In the following exercises, simplify each expression using the Product Property.

Try it.

ⓐ \({b}^{4}{b}^{-8}\) ⓑ \(({w}^{4}{x}^{-5})({w}^{-2}{x}^{-4})\) ⓒ \((-6{c}^{-3}{d}^{9})(2{c}^{4}{d}^{-5})\)

Solution

ⓐ \(\frac{1}{{b}^{4}}\) ⓑ \(\frac{{w}^{2}}{{x}^{9}}\) ⓒ \(-12c{d}^{4}\)

Try it.

ⓐ \({s}^{3}\cdot {s}^{-7}\) ⓑ \(({m}^{3}{n}^{-3})({m}^{-5}{n}^{-1})\) ⓒ \((-2{j}^{-5}{k}^{8})(7{j}^{2}{k}^{-3})\)

Try it.

ⓐ \({a}^{3}\cdot {a}^{-3}\) ⓑ \((u{v}^{-2})({u}^{-5}{v}^{-3})\) ⓒ \((-4{r}^{-2}{s}^{-8})(9{r}^{4}{s}^{3})\)

Solution

ⓐ 1 ⓑ \(\frac{1}{{u}^{4}{v}^{5}}\) ⓒ \(\frac{-36{r}^{2}}{{s}^{5}}\)

Try it.

ⓐ \({y}^{5}\cdot {y}^{-5}\) ⓑ \((p{q}^{-4})({p}^{-6}{q}^{-3})\) ⓒ \((-5{m}^{4}{n}^{6})(8{m}^{-5}{n}^{-3})\)

Try it.

\({p}^{5}\cdot {p}^{-2}\cdot {p}^{-4}\)

Solution

\(\frac{1}{p}\)

Try it.

\({x}^{4}\cdot {x}^{-2}\cdot {x}^{-3}\)

In the following exercises, simplify each expression using the Power Property.

Try it.

ⓐ \({({m}^{4})}^{2}\) ⓑ \({({10}^{3})}^{6}\) ⓒ \({({x}^{3})}^{-4}\)

Solution

ⓐ \({m}^{8}\) ⓑ \({10}^{18}\) ⓒ \(\frac{1}{{x}^{12}}\)

Try it.

ⓐ \({({b}^{2})}^{7}\) ⓑ \({({3}^{8})}^{2}\) ⓒ \({({k}^{2})}^{-5}\)

Try it.

ⓐ \({({y}^{3})}^{x}\) ⓑ \({({5}^{x})}^{y}\) ⓒ \({({q}^{6})}^{-8}\)

Solution

ⓐ \({y}^{3x}\) ⓑ \({5}^{xy}\) ⓒ \(\frac{1}{{q}^{48}}\)

Try it.

ⓐ \({({x}^{2})}^{y}\) ⓑ \({({7}^{a})}^{b}\) ⓒ \({({a}^{9})}^{-10}\)

In the following exercises, simplify each expression using the Product to a Power Property.

Try it.

ⓐ \({(-3xy)}^{2}\) ⓑ \({(6a)}^{0}\) ⓒ \({(5{x}^{2})}^{-2}\) ⓓ \({(-4{y}^{-3})}^{2}\)

Solution

ⓐ \(9{x}^{2}{y}^{2}\) ⓑ 1 ⓒ \(\frac{1}{25{x}^{4}}\)
ⓓ \(\frac{16}{{y}^{6}}\)

Try it.

ⓐ \({(-4ab)}^{2}\) ⓑ \({(5x)}^{0}\) ⓒ \({(4{y}^{3})}^{-3}\) ⓓ \({(-7{y}^{-3})}^{2}\)

Try it.

ⓐ \({(-5ab)}^{3}\) ⓑ \({(-4pq)}^{0}\) ⓒ \({(-6{x}^{3})}^{-2}\) ⓓ \({(3{y}^{-4})}^{2}\)

Solution

ⓐ \(-125{a}^{3}{b}^{3}\) ⓑ 1 ⓒ \(\frac{1}{36{x}^{6}}\) ⓓ \(\frac{9}{{y}^{8}}\)

Try it.

ⓐ \({(-3xyz)}^{4}\) ⓑ \({(-7mn)}^{0}\) ⓒ \({(-3{x}^{3})}^{-2}\)
ⓓ \({(2{y}^{-5})}^{2}\)

In the following exercises, simplify each expression using the Quotient to a Power Property.

Try it.

ⓐ \({(\frac{p}{2})}^{5}\) ⓑ \({(\frac{x}{y})}^{-6}\) ⓒ \({(\frac{2x{y}^{2}}{z})}^{3}\) ⓓ \({(\frac{4{p}^{-3}}{{q}^{2}})}^{2}\)

Solution

ⓐ \(\frac{{p}^{5}}{32}\) ⓑ \(\frac{{y}^{6}}{{x}^{6}}\) ⓒ \(\frac{8{x}^{3}{y}^{6}}{{z}^{3}}\)
ⓓ \(\frac{16}{{p}^{6}{q}^{4}}\)

Try it.

ⓐ \({(\frac{x}{3})}^{4}\) ⓑ \({(\frac{a}{b})}^{-5}\) ⓒ \({(\frac{2{x}^{2}{y}^{3}}{{z}^{2}})}^{2}\) ⓓ \({(\frac{{x}^{3}y}{{z}^{4}})}^{2}\)

Try it.

ⓐ \({(\frac{a}{3b})}^{4}\) ⓑ \({(\frac{5}{4m})}^{-2}\) ⓒ \({(\frac{3{a}^{-2}{b}^{3}}{{c}^{2}})}^{-2}\) ⓓ \({(\frac{{p}^{-1}{q}^{4}}{{r}^{-4}})}^{2}\)

Solution

ⓐ \(\frac{{a}^{4}}{81{b}^{4}}\) ⓑ \(\frac{16{m}^{2}}{25}\) ⓒ \(\frac{{a}^{4}{c}^{4}}{9{b}^{6}}\) ⓓ \(\frac{{q}^{8}{r}^{8}}{{p}^{2}}\)

Try it.

ⓐ \({(\frac{x}{2y})}^{3}\) ⓑ \({(\frac{10}{3q})}^{-4}\) ⓒ \({(\frac{2{x}^{3}{y}^{4}}{3{z}^{2}})}^{5}\) ⓓ \({(\frac{5{a}^{3}{b}^{-1}}{2{c}^{4}})}^{-3}\)

In the following exercises, simplify each expression by applying several properties.

Try it.

ⓐ \({(5{t}^{2})}^{3}{(3t)}^{2}\) ⓑ \(\frac{{({t}^{2})}^{5}{({t}^{-4})}^{2}}{{({t}^{3})}^{7}}\) ⓒ \({(\frac{2x{y}^{2}}{{x}^{3}{y}^{-2}})}^{2}{(\frac{12x{y}^{3}}{{x}^{3}{y}^{-1}})}^{-1}\)

Solution

ⓐ \(1125{t}^{8}\) ⓑ \(\frac{1}{{t}^{19}}\) ⓒ \(\frac{{y}^{4}}{3{x}^{2}}\)

Try it.

ⓐ \({(10{k}^{4})}^{3}{(5{k}^{6})}^{2}\) ⓑ \(\frac{{({q}^{3})}^{6}{({q}^{-2})}^{3}}{{({q}^{4})}^{8}}\)

Try it.

ⓐ \({({m}^{2}n)}^{2}{(2m{n}^{5})}^{4}\) ⓑ \(\frac{{(-2{p}^{-2})}^{4}{(3{p}^{4})}^{2}}{{(-6{p}^{3})}^{2}}\)

Solution

ⓐ \(16{m}^{8}{n}^{22}\) ⓑ \(\frac{4}{{p}^{6}}\)

Try it.

ⓐ \({(3p{q}^{4})}^{2}{(6{p}^{6}q)}^{2}\) ⓑ \(\frac{{(-2{k}^{-3})}^{2}{(6{k}^{2})}^{4}}{{(9{k}^{4})}^{2}}\)

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

1. Simplify: \((-2)(-2)(-2).\)
   If you missed this problem, review .

   Sýna svarið

   \(-8\)

2. Simplify: \(\frac{8x}{24y}.\)
   If you missed this problem, review .

   Sýna svarið

   \(\frac{x}{3y}\)

3. Name the decimal \((-2.6)(4.21).\)
   If you missed this problem, review .

   Sýna svarið

   \(-10.946\)

4. Simplify each expression: ⓐ \({y}^{5}\cdot {y}^{6}\) ⓑ \({2}^{x}\cdot {2}^{3x}\) ⓒ \(2{a}^{7}\cdot 3a.\) ⓓ \({d}^{4}⋅{d}^{5}⋅{d}^{2}\)

   Sýna svarið

   ⓐ
   

   Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
   Simplify.

   ⓑ
   

   Use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
   Simplify.

   ⓒ
   

   Rewrite, \(a={a}^{1}.\)
   Use the Commutative Property and use the Product Property, \({a}^{m}\cdot {a}^{n}={a}^{m+n}.\)
   Simplify.

   ⓓ
   

   Add the exponents, since bases are the same.
   Simplify.

5. Simplify each expression:

   ⓐ \({b}^{9}\cdot {b}^{8}\) ⓑ \({4}^{2x}\cdot {4}^{x}\) ⓒ \(3{p}^{5}\cdot 4p\) ⓓ \({x}^{6}\cdot {x}^{4}\cdot {x}^{8}.\)

   Sýna svarið

   ⓐ \({b}^{17}\) ⓑ \({4}^{3x}\) ⓒ \(12{p}^{6}\)
   ⓓ \({x}^{18}\)

6. Simplify each expression:

   ⓐ \({x}^{12}\cdot {x}^{4}\) ⓑ \(10\cdot {10}^{x}\) ⓒ \(2z\cdot 6{z}^{7}\) ⓓ \({b}^{5}\cdot {b}^{9}\cdot {b}^{5}.\)

   Sýna svarið

   ⓐ \({x}^{16}\) ⓑ \({10}^{x+1}\) ⓐ \(12{z}^{8}\)
   ⓓ \({b}^{19}\)

7. Simplify each expression: ⓐ \(\frac{{x}^{9}}{{x}^{7}}\) ⓑ \(\frac{{3}^{10}}{{3}^{2}}\) ⓒ \(\frac{{b}^{8}}{{b}^{12}}\) ⓓ \(\frac{{7}^{3}}{{7}^{5}}.\)

   Sýna svarið

   To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

   ⓐ
   

   Since \(9>7,\) there are more factors of \(x\) in the numerator.\(\\)
   Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)
   Simplify.

   ⓑ
   

   Since \(10>2,\) there are more factors of \(3\) in the numerator.\(\\)
   Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)
   Simplify.

   Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

   ⓒ
   

   Since \(12>8,\) there are more factors of \(b\) in the denominator.
   Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
   Simplify.

   ⓓ
   

   Since \(5>3,\) there are more factors of \(3\) in the denominator.\(\\)
   Use Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
   Simplify.
   Simplify.

   Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

8. Simplify each expression: ⓐ \(\frac{{x}^{15}}{{x}^{10}}\) ⓑ \(\frac{{6}^{14}}{{6}^{5}}\) ⓒ \(\frac{{x}^{18}}{{x}^{22}}\) ⓓ \(\frac{{12}^{15}}{{12}^{30}}.\)

   Sýna svarið

   ⓐ \({x}^{5}\) ⓑ \({6}^{9}\) ⓒ \(\frac{1}{{x}^{4}}\)
   ⓓ \(\frac{1}{{12}^{15}}\)

9. Simplify each expression: ⓐ \(\frac{{y}^{43}}{{y}^{37}}\) ⓑ \(\frac{{10}^{15}}{{10}^{7}}\) ⓒ \(\frac{{m}^{7}}{{m}^{15}}\) ⓓ \(\frac{{9}^{8}}{{9}^{19}}.\)

   Sýna svarið

   ⓐ \({y}^{6}\) ⓑ \({10}^{8}\) ⓒ \(\frac{1}{{m}^{8}}\)
   ⓓ \(\frac{1}{{9}^{11}}\)

10. Simplify each expression: ⓐ \({9}^{0}\) ⓑ \({n}^{0}.\)

    Sýna svarið

    The definition says any non-zero number raised to the zero power is 1.

    ⓐ
    \(\begin{array}{llll} & & & \ {9}^{0} \\ \text{Use the definition of the zero exponent.} & & & \ 1\end{array}\)

    ⓑ
    \(\begin{array}{llll} & & & \ {n}^{0} \\ \text{Use the definition of the zero exponent.} & & & \ 1\end{array}\)

    To simplify the expression n raised to the zero power we just use the definition of the zero exponent. The result is 1.

11. Simplify each expression: ⓐ \({11}^{0}\) ⓑ \({q}^{0}.\)

    Sýna svarið

    ⓐ 1 ⓑ 1

12. Simplify each expression: ⓐ \({23}^{0}\) ⓑ \({r}^{0}.\)

    Sýna svarið

    ⓐ 1 ⓑ 1

13. Simplify each expression: ⓐ \({x}^{-5}\) ⓑ \({10}^{-3}\) ⓒ \(\frac{1}{{y}^{-4}}\) ⓓ \(\frac{1}{{3}^{-2}}.\)

    Sýna svarið

    ⓐ
    

    	\(\ {x}^{-5}\)
    Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{{x}^{5}}\)

    ⓑ
    

    	\(\ {10}^{-3}\)
    Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ \frac{1}{{10}^{3}}\)
    Simplify.	\(\ \frac{1}{1000}\)

    ⓒ
    

    	\(\ \frac{1}{{y}^{-4}}\)
    Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}.\)	\(\ {y}^{4}\)

    ⓓ
    

    	\(\ \frac{1}{{3}^{-2}}\)
    Use the property of a negative exponent, \(\frac{1}{{a}^{\text{-}n}}={a}^{n}.\)	\(\ {3}^{2}\)
    Simplify.	\(\ 9\)

14. Simplify each expression: ⓐ \({z}^{-3}\) ⓑ \({10}^{-7}\) ⓒ \(\frac{1}{{p}^{-8}}\) ⓓ \(\frac{1}{{4}^{-3}}.\)

    Sýna svarið

    ⓐ \(\frac{1}{{z}^{3}}\) ⓑ \(\frac{1}{{10}^{7}}\) ⓒ \({p}^{8}\) ⓓ \(64\)

15. Simplify each expression: ⓐ \({n}^{-2}\) ⓑ \({10}^{-4}\) ⓒ \(\frac{1}{{q}^{-7}}\) ⓓ \(\frac{1}{{2}^{-4}}.\)

    Sýna svarið

    ⓐ \(\frac{1}{{n}^{2}}\) ⓑ \(\frac{1}{10,000}\) ⓒ \({q}^{7}\)
    ⓓ \(16\)

16. Simplify each expression: ⓐ \({(\frac{5}{7})}^{-2}\) ⓑ \({(-\frac{x}{y})}^{-3}.\)

    Sýna svarið

    ⓐ
    

    	\(\ {(\frac{5}{7})}^{-2}\)
    Use the Quotient to a Negative Exponent Property, \({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}.\) Take the reciprocal of the fraction and change the sign of the exponent.	\(\ {(\frac{7}{5})}^{2}\)
    Simplify.	\(\ \frac{49}{25}\)

    ⓑ
    

    	\(\ {(-\frac{x}{y})}^{-3}\)
    Use the Quotient to a Negative Exponent Property, \({(\frac{a}{b})}^{\text{-}n}={(\frac{b}{a})}^{n}.\) Take the reciprocal of the fraction and change the sign of the exponent.	\(\ {(-\frac{y}{x})}^{3}\)
    Simplify.	\(\ -\frac{{y}^{3}}{{x}^{3}}\)

17. Simplify each expression: ⓐ \({(\frac{2}{3})}^{-4}\) ⓑ \({(-\frac{m}{n})}^{-2}.\)

    Sýna svarið

    ⓐ \(\frac{81}{16}\) ⓑ \(\frac{{n}^{2}}{{m}^{2}}\)

18. Simplify each expression: ⓐ \({(\frac{3}{5})}^{-3}\) ⓑ \({(-\frac{a}{b})}^{-4}.\)

    Sýna svarið

    ⓐ \(\frac{125}{27}\) ⓑ \(\frac{{b}^{4}}{{a}^{4}}\)

19. Simplify each expression: ⓐ \({z}^{-5}\cdot {z}^{-3}\) ⓑ \(({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\) ⓒ \((2{x}^{-6}{y}^{8})(-5{x}^{5}{y}^{-3}).\)

    Sýna svarið

    ⓐ
    

    	\(\ {z}^{-5}\cdot {z}^{-3}\)
    Add the exponents, since the bases are the same.	\(\ {z}^{-5-3}\)
    Simplify.	\(\ {z}^{-8}\)
    Use the definition of a negative exponent.	\(\ \frac{1}{{z}^{8}}\)

    ⓑ
    

    	\(\ ({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})\)
    Use the Commutative Property to get like bases together.	\(\ {m}^{4}{m}^{-5}\cdot {n}^{-2}{n}^{-3}\)
    Add the exponents for each base.	\(\ {m}^{-1}\cdot {n}^{-5}\)
    Take reciprocals and change the signs of the exponents.	\(\ \frac{1}{{m}^{1}}\cdot \frac{1}{{n}^{5}}\)
    Simplify.	\(\ \frac{1}{m{n}^{5}}\)

    ⓒ
    

    	\(\ (2{x}^{-6}{y}^{8})(-5{x}^{5}{y}^{-3})\)
    Rewrite with the like bases together.	\(\ 2(-5)\cdot ({x}^{-6}{x}^{5})\cdot ({y}^{8}{y}^{-3})\)
    Multiply the coefficients and add the exponents of each variable.	\(\ -10\cdot {x}^{-1}\cdot {y}^{5}\)
    Use the definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\ -10\cdot \frac{1}{x}\cdot {y}^{5}\)
    Simplify.	\(\ \frac{-10{y}^{5}}{x}\)

20. Simplify each expression:

    ⓐ \({z}^{-4}\cdot {z}^{-5}\) ⓑ \(({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1})\) ⓒ \((3{u}^{-5}{v}^{7})(-4{u}^{4}{v}^{-2}).\)

    Sýna svarið

    ⓐ \(\frac{1}{{z}^{9}}\) ⓑ \(\frac{1}{{p}^{3}{q}^{3}}\) ⓒ \(-\frac{12{v}^{5}}{u}\)

21. Simplify each expression:

    ⓐ \({c}^{-8}\cdot {c}^{-7}\) ⓑ \(({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5})\) ⓒ \((-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).\)

    Sýna svarið

    ⓐ \(\frac{1}{{c}^{15}}\) ⓑ \(\frac{1}{{r}^{2}{s}^{8}}\) ⓒ \(\frac{30{d}^{3}}{{c}^{8}}\)

22. Simplify each expression: ⓐ \({({y}^{5})}^{9}\) ⓑ \({({4}^{4})}^{7}\) ⓒ \({({y}^{3})}^{6}{({y}^{5})}^{4}.\)

    Sýna svarið

    ⓐ
    

    Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\\)
    Simplify.

    ⓑ
    

    Use the Power Property.
    Simplify.

    ⓒ
    

    	\(\ {({y}^{3})}^{6}{({y}^{5})}^{4}\)
    Use the Power Property.	\(\ {y}^{18}\cdot {y}^{20}\)
    Add the exponents.	\(\ {y}^{38}\)

23. Simplify each expression: ⓐ \({({b}^{7})}^{5}\) ⓑ \({({5}^{4})}^{3}\) ⓒ \({({a}^{4})}^{5}{({a}^{7})}^{4}.\)

    Sýna svarið

    ⓐ \({b}^{35}\) ⓑ \({5}^{12}\) ⓒ \({a}^{48}\)

24. Simplify each expression: ⓐ \({({z}^{6})}^{9}\) ⓑ \({({3}^{7})}^{7}\) ⓒ \({({q}^{4})}^{5}{({q}^{3})}^{3}.\)

    Sýna svarið

    ⓐ \({z}^{54}\) ⓑ \({3}^{49}\) ⓒ \({q}^{29}\)

25. Simplify each expression: ⓐ \({(-3mn)}^{3}\) ⓑ \({(-4{a}^{2}b)}^{0}\) ⓒ \({(6{k}^{3})}^{-2}\) ⓓ \({(5{x}^{-3})}^{2}.\)

    Sýna svarið

    ⓐ
    

    Use Power of a Product Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)
    Simplify.

    ⓑ
    

    	\(\ {(-4{a}^{2}b)}^{0}\)
    Use Power of a Product Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\ {(-4)}^{0}{({a}^{2})}^{0}{(b)}^{0}\)
    Simplify.	\(\ 1\cdot 1\cdot 1\)
    Multiply.	\(\ 1\)

    ⓒ
    

    	\({(6{k}^{3})}^{-2}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\({(6)}^{-2}{({k}^{3})}^{-2}\)
    Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)	\({6}^{-2}{k}^{-6}\)
    Use the Definition of a negative exponent, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(\frac{1}{{6}^{2}}\cdot \frac{1}{{k}^{6}}\)
    Simplify.	\(\frac{1}{36{k}^{6}}\)

    ⓓ
    

    	\({(5{x}^{-3})}^{2}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\({5}^{2}{({x}^{-3})}^{2}\)
    Simplify.	\(25\cdot {x}^{-6}\)
    Rewrite \({x}^{-6}\) using, \({a}^{\text{-}n}=\frac{1}{{a}^{n}}.\)	\(25\cdot \frac{1}{{x}^{6}}\)
    Simplify.	\(\frac{25}{{x}^{6}}\)

26. Simplify each expression: ⓐ \({(2wx)}^{5}\) ⓑ \({(-11p{q}^{3})}^{0}\) ⓒ \({(2{b}^{3})}^{-4}\) ⓓ \({(8{a}^{-4})}^{2}.\)

    Sýna svarið

    ⓐ \(32{w}^{5}{x}^{5}\) ⓑ 1 ⓒ \(\frac{1}{16{b}^{12}}\)
    ⓓ \(\frac{64}{{a}^{8}}\)

27. Simplify each expression: ⓐ \({(-3y)}^{3}\) ⓑ \({(-8{m}^{2}{n}^{3})}^{0}\) ⓒ \({(-4{x}^{4})}^{-2}\) ⓓ \({(2{c}^{-4})}^{3}.\)

    Sýna svarið

    ⓐ \(-27{y}^{3}\) ⓑ 1 ⓒ \(\frac{1}{16{x}^{8}}\)
    ⓓ \(\frac{8}{{c}^{12}}\)

28. Simplify each expression:

    ⓐ \({(\frac{b}{3})}^{4}\) ⓑ \({(\frac{k}{j})}^{-3}\) ⓒ \({(\frac{2x{y}^{2}}{z})}^{3}\) ⓓ \({(\frac{4{p}^{-3}}{{q}^{2}})}^{2}.\)

    Sýna svarið

    ⓐ
    

    Use Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)
    Simplify.

    ⓑ
    

    Raise the numerator and denominator to the power.
    Use the definition of negative exponent.
    Multiply.

    ⓒ
    

    	\({(\frac{2x{y}^{2}}{z})}^{3}\)
    Use Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)	\(\frac{{(2x{y}^{2})}^{3}}{{z}^{3}}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\frac{8{x}^{3}{y}^{6}}{{z}^{3}}\)

    ⓓ
    

    	\({(\frac{4{p}^{-3}}{{q}^{2}})}^{2}\)
    Use Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)	\(\frac{{(4{p}^{-3})}^{2}}{{({q}^{2})}^{2}}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\frac{{4}^{2}{({p}^{-3})}^{2}}{{({q}^{2})}^{2}}\)
    Simplify using the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)	\(\frac{16{p}^{-6}}{{q}^{4}}\)
    Use the definition of negative exponent.	\(\frac{16}{{q}^{4}}\cdot \frac{1}{{p}^{6}}\)
    Simplify.	\(\frac{16}{{p}^{6}{q}^{4}}\)

29. Simplify each expression:

    ⓐ \({(\frac{p}{10})}^{4}\) ⓑ \({(\frac{m}{n})}^{-7}\) ⓒ \({(\frac{3a{b}^{3}}{{c}^{2}})}^{4}\) ⓓ \({(\frac{3{x}^{-2}}{{y}^{3}})}^{3}.\)

    Sýna svarið

    ⓐ \(\frac{{p}^{4}}{10000}\) ⓑ \(\frac{{n}^{7}}{{m}^{7}}\)
    ⓒ \(\frac{81{a}^{4}{b}^{12}}{{c}^{8}}\) ⓓ \(\frac{27}{{x}^{6}{y}^{9}}\)

30. Simplify each expression:

    ⓐ \({(\frac{-2}{q})}^{3}\) ⓑ \({(\frac{w}{x})}^{-4}\) ⓒ \({(\frac{x{y}^{3}}{3{z}^{2}})}^{2}\) ⓓ \({(\frac{2{m}^{-2}}{{n}^{-2}})}^{3}.\)

    Sýna svarið

    ⓐ \(\frac{-8}{{q}^{3}}\) ⓑ \(\frac{{x}^{4}}{{w}^{4}}\) ⓒ \(\frac{{x}^{2}{y}^{6}}{9{z}^{4}}\)
    ⓓ \(\frac{8{n}^{6}}{{m}^{6}}\)

31. Simplify each expression by applying several properties:

    ⓐ \({(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}\) ⓑ \(\frac{{({x}^{3})}^{4}{({x}^{-2})}^{5}}{{({x}^{6})}^{5}}\) ⓒ \({(\frac{2x{y}^{2}}{{x}^{3}{y}^{-2}})}^{2}{(\frac{12x{y}^{3}}{{x}^{3}{y}^{-1}})}^{-1}.\)

    Sýna svarið

    ⓐ
    

    	\({(3{x}^{2}y)}^{4}{(2x{y}^{2})}^{3}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(({3}^{4}{x}^{8}{y}^{4})({2}^{3}{x}^{3}{y}^{6})\)
    Simplify.	\((81{x}^{8}{y}^{4})(8{x}^{3}{y}^{6})\)
    Use the Commutative Property.	\(81\cdot 8\cdot {x}^{8}\cdot {x}^{3}\cdot {y}^{4}\cdot {y}^{6}\)
    Multiply the constants and add the exponents.	\(648{x}^{11}{y}^{10}\)

    ⓑ
    

    	\(\ \frac{{({x}^{3})}^{4}{({x}^{-2})}^{5}}{{({x}^{6})}^{5}}\)
    Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}.\)	\(\ \frac{({x}^{12})({x}^{-10})}{({x}^{30})}\)
    Add the exponents in the numerator.	\(\ \frac{{x}^{2}}{{x}^{30}}\)
    Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)	\(\ \frac{1}{{x}^{28}}\)

    ⓒ
    

    	\({(\frac{2x{y}^{2}}{{x}^{3}{y}^{-2}})}^{2}{(\frac{12x{y}^{3}}{{x}^{3}{y}^{-1}})}^{-1}\)
    Simplify inside the parentheses first.	\({(\frac{2{y}^{4}}{{x}^{2}})}^{2}{(\frac{12{y}^{4}}{{x}^{2}})}^{-1}\)
    Use the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}.\)	\(\frac{{(2{y}^{4})}^{2}}{{({x}^{2})}^{2}}\frac{{(12{y}^{4})}^{-1}}{{({x}^{2})}^{-1}}\)
    Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}.\)	\(\frac{4{y}^{8}}{{x}^{4}}\cdot \frac{{12}^{-1}{y}^{-4}}{{x}^{-2}}\)
    Simplify.	\(\frac{4{y}^{4}}{12{x}^{2}}\)
    Simplify.	\(\frac{{y}^{4}}{3{x}^{2}}\)

32. Simplify each expression:

    ⓐ \({({c}^{4}{d}^{2})}^{5}{(3c{d}^{5})}^{4}\) ⓑ \(\frac{{({a}^{-2})}^{3}{({a}^{2})}^{4}}{{({a}^{4})}^{5}}\) ⓒ \({(\frac{3x{y}^{2}}{{x}^{2}{y}^{-3}})}^{2}{(\frac{9x{y}^{-3}}{{x}^{3}{y}^{2}})}^{-1}.\)

    Sýna svarið

    ⓐ \(81{c}^{24}{d}^{30}\) ⓑ \(\frac{1}{{a}^{18}}\)
    ⓒ \({y}^{15}\)

33. Simplify each expression:

    ⓐ \({({a}^{3}{b}^{2})}^{6}{(4a{b}^{3})}^{4}\) ⓑ \(\frac{{({p}^{-3})}^{4}{({p}^{5})}^{3}}{{({p}^{7})}^{6}}\) ⓒ \({(\frac{4{x}^{3}{y}^{2}}{{x}^{2}{y}^{-1}})}^{2}{(\frac{8x{y}^{-3}}{{x}^{2}{y}^{}})}^{-1}.\)

    Sýna svarið

    ⓐ \(256{a}^{22}{b}^{24}\) ⓑ \(\frac{1}{{p}^{39}}\)
    ⓒ \(2{x}^{3}{y}^{10}\)

34. Write in scientific notation: ⓐ 37,000 ⓑ \(0.0052.\)

    Sýna svarið

    ⓐ
    

    The original number, 37,000, is greater than 1 so we will have a positive power of 10.	\(\\)37,000
    Move the decimal point to get 3.7, a number between 1 and 10.
    Count the number of decimal places the point was moved.
    Write as a product with a power of 10.
    Check:    \(\begin{array}{l} \\ 3.7\ \times \ {10}^{4} \\ 3.7\ \times \ 10,000 \\ 37,000\end{array}\)

    ⓑ
    

    The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10.	\(\\)0.0052
    Move the decimal point to get 5.2, a number between 1 and 10.
    Count the number of decimal places the point was moved.
    Write as a product with a power of 10.
    \(\begin{array}{llll} \\ \\ \text{Check:} & & & 5.2\ \times \ {10}^{-3} \\ & & & 5.2\ \times \ \frac{1}{{10}^{3}} \\ & & & 5.2\ \times \ \frac{1}{1000} \\ & & & 5.2\ \times \ 0.001 \\ & & & 0.0052\end{array}\)

35. Write in scientific notation: ⓐ 96,000 ⓑ 0.0078.

    Sýna svarið

    ⓐ \(9.6\ \times \ {10}^{4}\) ⓑ \(7.8\ \times \ {10}^{-3}\)

36. Write in scientific notation: ⓐ 48,300 ⓑ 0.0129.

    Sýna svarið

    ⓐ \(4.83\ \times \ {10}^{4}\)
    ⓑ \(1.29\ \times \ {10}^{-2}\)

37. Convert to decimal form: ⓐ \(6.2\ \times \ {10}^{3}\) ⓑ \(-8.9\ \times \ {10}^{-2}.\)

    Sýna svarið

    ⓐ
    

    Determine the exponent, n, on the factor 10.
    	The exponent is 3.
    Since the exponent is positive, move the decimal point 3 places to the right.
    Add zeros as needed for placeholders.

    ⓑ
    

    Determine the exponent, n, on the factor 10.	The exponent is \(-2.\)
    Since the exponent is negative, move the decimal point 2 places to the left.
    Add zeros as needed for placeholders.

38. Convert to decimal form: ⓐ \(1.3\ \times \ {10}^{3}\) ⓑ \(-1.2\ \times \ {10}^{-4}.\)

    Sýna svarið

    ⓐ 1,300 ⓑ \(-0.00012\)

39. Convert to decimal form: ⓐ \(-9.5\ \times \ {10}^{4}\) ⓑ \(7.5\ \times \ {10}^{-2}.\)

    Sýna svarið

    ⓐ \(-950,000\) ⓑ 0.075

40. Multiply or divide as indicated. Write answers in decimal form: ⓐ \((-4\ \times \ {10}^{5})(2\ \times \ {10}^{-7})\) ⓑ \(\frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}.\)

    Sýna svarið

    ⓐ
    

    	\((-4\ \times \ {10}^{5})(2\ \times \ {10}^{-7})\)
    Use the Commutative Property to rearrange the factors.	\(-4\cdot 2\cdot {10}^{5}\cdot {10}^{-7}\)
    Multiply.	\(-8\ \times \ {10}^{-2}\)
    Change to decimal form by moving the decimal two places left.	\(-0.08\)

    ⓑ
    

    	\(\ \frac{9\ \times \ {10}^{3}}{3\ \times \ {10}^{-2}}\)
    Separate the factors, rewriting as the product of two fractions.	\(\ \frac{9}{3}\ \times \ \frac{{10}^{3}}{{10}^{-2}}\)
    Divide.	\(\ 3\ \times \ {10}^{5}\)
    Change to decimal form by moving the decimal five places right.	\(\ 300,000\)

Symbols used here

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

i

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Properties of Exponents and Scientific Notation

1. Simplify expressions using the properties for exponents
2. Use the definition of a negative exponent
3. Use scientific notation
4. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
5. Count the number of decimal places,
6. Write the number as a product with a power of 10. If the original number is.
7. greater than 1, the power of 10 will be
8. between 0 and 1, the power of 10 will be

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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