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Simplify Expressions with Roots

Simplify expressions with roots

Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write \({n}^{2}\) and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,

\[\begin{array}{l}{13}^{2}\ \text{is read “13 squared”} \\ \text{169 is called the}\ \text{square}\ \text{of 13, since}\ {13}^{2}=169 \\ \text{13 is a}\ \text{square root}\ \text{of 169}\end{array}\]

Notice (−13)² = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, \(\sqrt{m},\) which denotes the positive square root of m. The positive square root is also called the principal square root. This symbol, as well as other radicals to be introduced later, are grouping symbols.

We also use the radical sign for the square root of zero. Because \({0}^{2}=0,\) \(\sqrt{0}=0.\) Notice that zero has only one square root.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write \(\sqrt{169}=13.\) If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, \(\text{-}\sqrt{169}=-13.\)

Example

Try it.

Simplify: ⓐ \(\sqrt{144}\) ⓑ \(\text{-}\sqrt{289}.\)

Solution

ⓐ

	\(\sqrt{144}\)
Since \({12}^{2}=144.\)	\(12\)

ⓑ

	\(-\sqrt{289}\)
Since \({17}^{2}=289\) and the negative is in front of the radical sign.	\(-17\)

Can we simplify \(\sqrt{-49}?\) Is there a number whose square is \(-49?\)

\[{(\ )}^{2}=-49\]

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to \(\sqrt{-49}.\) The square root of a negative number is not a real number.

Example

Try it.

Simplify: ⓐ \(\sqrt{-196}\) ⓑ \(\text{-}\sqrt{64}.\)

Solution

ⓐ

	\(\sqrt{-196}\)
There is no real number whose square is \(-196.\)	\(\sqrt{-196}\ \text{is not a real number.}\)

ⓑ

	\(-\sqrt{64}\)
The negative is in front of the radical.	\(-8\)

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

Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the \(\sqrt{4}=2,\) what is the value of \(\sqrt[]{21}\) or \(\sqrt[3]{50}?\) In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of \(\sqrt{11},\) we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.

Similarly, to estimate \(\sqrt[3]{91},\) we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.

Example

Try it.

Estimate each root between two consecutive whole numbers: ⓐ \(\sqrt{105}\) ⓑ \(\sqrt[3]{43}.\)

Solution

ⓐ Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.



Locate 105 between two consecutive perfect squares.
\(\sqrt{105}\) is between their square roots.

ⓑ Similarly we locate 43 between two perfect cube numbers.



Locate 43 between two consecutive perfect cubes.
\(\sqrt[3]{43}\) is between their cube roots.

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the \(\sqrt{x}\) key on your calculator. To find a cube root, or any root with higher index, you will use the \(\sqrt[y]{x}\) key.

When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is \(\approx\) and it is read ‘approximately’.

Suppose your calculator has a 10 digit display. You would see that

\[\begin{array}{l}\sqrt{5}\approx 2.236067978\ \text{rounded to two decimal places is}\ \sqrt{5}\approx 2.24 \\ \sqrt[4]{93}\approx 3.105422799\ \text{rounded to two decimal places is}\ \sqrt[4]{93}\approx 3.11\end{array}\]

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

\[\begin{array}{lll}{(2.236067978)}^{2} & = & 5.000000002 \\ {(2.24)}^{2} & = & 5.0176\end{array}\ \begin{array}{lll}{(3.105422799)}^{4} & = & 92.999999991 \\ {(3.11)}^{4} & = & 93.54951841\end{array}\]

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

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

But what about an even root? We want the principal root, so \(\sqrt[4]{625}=5.\)

But notice,

How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value. \(|-5|=5.\) So we say that when n is even \(\sqrt[n]{{a}^{n}}=|a|.\) This guarantees the principal root is positive.

Example

Try it.

Simplify: ⓐ \(\sqrt{{x}^{2}}\) ⓑ \(\sqrt[3]{{n}^{3}}\) ⓒ \(\sqrt[4]{{p}^{4}}\) ⓓ \(\sqrt[5]{{y}^{5}}.\)

Solution

ⓐ We use the absolute value to be sure to get the positive root.

	\(\sqrt{{x}^{2}}\)
Since the index \(n\) is even, \(\sqrt[n]{{a}^{n}}=\|a\|.\)	\(\|x\|\)

ⓑ This is an odd indexed root so there is no need for an absolute value sign.

	\(\sqrt[3]{{n}^{3}}\)
Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\)	\(n\)

ⓒ

	\(\sqrt[4]{{p}^{4}}\)
Since the index \(n\ \text{is even}\ \sqrt[n]{{a}^{n}}=\|a\|.\)	\(\|p\|\)

ⓓ

	\(\sqrt[5]{{y}^{5}}\)
Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\)	\(y\)

What about square roots of higher powers of variables? The Power Property of Exponents says \({({a}^{m})}^{n}={a}^{m\cdot n}.\) So if we square a^m, the exponent will become 2m.

\[{({a}^{m})}^{2}={a}^{2m}\]

Looking now at the square root,

\(\begin{array}{llll} & & & \ \sqrt{{a}^{2m}} \\ \text{Since}\ {({a}^{m})}^{2}={a}^{2m}. & & & \ \sqrt{{({a}^{m})}^{2}} \\ \text{Since}\ n\ \text{is even}\ \sqrt[n]{{a}^{n}}=|a|. & & & \ |{a}^{m}| \\ & & & \ \text{So}\ \sqrt{{a}^{2m}}=|{a}^{m}|.\end{array}\)

Example

Try it.

Simplify: ⓐ \(\sqrt{{x}^{6}}\) ⓑ \(\sqrt{{y}^{16}}.\)

Solution

ⓐ

	\(\sqrt{{x}^{6}}\)
Since \({({x}^{3})}^{2}={x}^{6}.\)	\(\sqrt{{({x}^{3})}^{2}}\)
Since the index \(n\) is even \(\sqrt{{a}^{n}}=\|a\|.\)	\(\|{x}^{3}\|\)

ⓑ

	\(\sqrt{{y}^{16}}\)
Since \({({y}^{8})}^{2}={y}^{16}.\)	\(\sqrt{{({y}^{8})}^{2}}\)
Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=\|a\|.\)	\({y}^{8}\)
In this case the absolute value sign is not needed as \({y}^{8}\) is positive.

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

Key Concepts

Square Root Notation
- \(\sqrt{m}\) is read ‘the square root of m’
- If n² = m, then \(n=\sqrt{m},\) for \(n\ge 0.\)
- The square root of m, \(\sqrt{m},\) is a positive number whose square is m.
n^th Root of a Number
- If \({b}^{n}=a,\) then b is an n^th root of a.
- The principal n^th root of a is written \(\sqrt[n]{a}.\)
- n is called the index of the radical.
Properties of \(\sqrt[n]{a}\)
- When n is an even number and
  - \(a\ge 0,\) then \(\sqrt[n]{a}\) is a real number
  - \(a<0,\) then \(\sqrt[n]{a}\) is not a real number
- When n is an odd number, \(\sqrt[n]{a}\) is a real number for all values of a.
Simplifying Odd and Even Roots
- For any integer \(n\ge 2,\)
  - when n is odd \(\sqrt[n]{{a}^{n}}=a\)
  - when n is even \(\sqrt[n]{{a}^{n}}=|a|\)
- We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Simplify Expressions with Roots

Simplify Expressions with Roots

In the following exercises, simplify.

Try it.

ⓐ \(\sqrt{64}\) ⓑ \(\text{-}\sqrt{81}\)

Solution

ⓐ 8 ⓑ \(-9\)

Try it.

ⓐ \(\sqrt{169}\) ⓑ \(\text{-}\sqrt{100}\)

Try it.

ⓐ \(\sqrt{196}\) ⓑ \(\text{-}\sqrt{1}\)

Solution

ⓐ 14 ⓑ \(-1\)

Try it.

ⓐ \(\sqrt{144}\) ⓑ \(\text{-}\sqrt{121}\)

Try it.

ⓐ \(\sqrt{\frac{4}{9}}\) ⓑ \(\text{-}\sqrt{0.01}\)

Solution

ⓐ \(\frac{2}{3}\) ⓑ \(-0.1\)

Try it.

ⓐ \(\sqrt{\frac{64}{121}}\) ⓑ \(\text{-}\sqrt{0.16}\)

Try it.

ⓐ \(\sqrt{-121}\) ⓑ \(\text{-}\sqrt{289}\)

Solution

ⓐ not real number ⓑ \(-17\)

Try it.

ⓐ \(\text{-}\sqrt{400}\) ⓑ \(\sqrt{-36}\)

Try it.

ⓐ \(\text{-}\sqrt{225}\) ⓑ \(\sqrt{-9}\)

Solution

ⓐ \(-15\) ⓑ not real number

Try it.

ⓐ \(\sqrt{-49}\) ⓑ \(\text{-}\sqrt{256}\)

Try it.

ⓐ \(\sqrt[3]{216}\) ⓑ \(\sqrt[4]{256}\)

Solution

ⓐ 6 ⓑ 4

Try it.

ⓐ \(\sqrt[3]{27}\) ⓑ \(\sqrt[4]{16}\) ⓒ \(\sqrt[5]{243}\)

Try it.

ⓐ \(\sqrt[3]{512}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{1}\)

Solution

ⓐ 8 ⓑ 3 ⓒ 1

Try it.

ⓐ \(\sqrt[3]{125}\) ⓑ \(\sqrt[4]{1296}\) ⓒ \(\sqrt[5]{1024}\)

Try it.

ⓐ \(\sqrt[3]{-8}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-32}\)

Solution

ⓐ \(-2\) ⓑ \(\text{not real}\) ⓒ \(-2\)

Try it.

ⓐ \(\sqrt[3]{-64}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}\)

Try it.

ⓐ \(\sqrt[3]{-125}\) ⓑ \(\sqrt[4]{-1296}\) ⓒ \(\sqrt[5]{-1024}\)

Solution

ⓐ \(-5\) ⓑ \(\text{not real}\) ⓒ \(-4\)

Try it.

ⓐ \(\sqrt[3]{-512}\) ⓑ \(\sqrt[4]{-81}\) ⓒ \(\sqrt[5]{-1}\)

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

Try it.

ⓐ \(\sqrt{70}\) ⓑ \(\sqrt[3]{71}\)

Solution

ⓐ \(8<\sqrt{70}<9\)
ⓑ \(4<\sqrt[3]{71}<5\)

Try it.

ⓐ \(\sqrt{55}\) ⓑ \(\sqrt[3]{119}\)

Try it.

ⓐ \(\sqrt{200}\) ⓑ \(\sqrt[3]{137}\)

Solution

ⓐ \(14<\sqrt{200}<15\)
ⓑ \(5<\sqrt[3]{137}<6\)

Try it.

ⓐ \(\sqrt{172}\) ⓑ \(\sqrt[3]{200}\)

In the following exercises, approximate each root and round to two decimal places.

Try it.

ⓐ \(\sqrt{19}\) ⓑ \(\sqrt[3]{89}\) ⓒ \(\sqrt[4]{97}\)

Solution

ⓐ \(\approx 4.36\) ⓑ \(\approx 4.46\)
ⓒ \(\approx 3.14\)

Try it.

ⓐ \(\sqrt{21}\) ⓑ \(\sqrt[3]{93}\) ⓒ \(\sqrt[4]{101}\)

Try it.

ⓐ \(\sqrt{53}\) ⓑ \(\sqrt[3]{147}\) ⓒ \(\sqrt[4]{452}\)

Solution

ⓐ \(\approx 7.28\) ⓑ \(\approx 5.28\)
ⓒ \(\approx 4.61\)

Try it.

ⓐ \(\sqrt{47}\) ⓑ \(\sqrt[3]{163}\) ⓒ \(\sqrt[4]{527}\)

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

Try it.

ⓐ \(\sqrt[5]{{u}^{5}}\) ⓑ \(\sqrt[8]{{v}^{8}}\)

Solution

ⓐ u ⓑ \(|v|\)

Try it.

ⓐ \(\sqrt[3]{{a}^{3}}\) ⓑ \(\sqrt[9]{{b}^{9}}\)

Try it.

ⓐ \(\sqrt[4]{{y}^{4}}\) ⓑ \(\sqrt[7]{{m}^{7}}\)

Solution

ⓐ \(|y|\) ⓑ \(m\)

Try it.

ⓐ \(\sqrt[8]{{k}^{8}}\) ⓑ \(\sqrt[6]{{p}^{6}}\)

Try it.

ⓐ \(\sqrt{{x}^{6}}\) ⓑ \(\sqrt{{y}^{16}}\)

Solution

ⓐ \(|{x}^{3}|\) ⓑ \({y}^{8}\)

Try it.

ⓐ \(\sqrt{{a}^{14}}\) ⓑ \(\sqrt{{w}^{24}}\)

Try it.

ⓐ \(\sqrt{{x}^{24}}\) ⓑ \(\sqrt{{y}^{22}}\)

Solution

ⓐ \({x}^{12}\) ⓑ \(|{y}^{11}|\)

Try it.

ⓐ \(\sqrt{{a}^{12}}\) ⓑ \(\sqrt{{b}^{26}}\)

Try it.

ⓐ \(\sqrt[3]{{x}^{9}}\) ⓑ \(\sqrt[4]{{y}^{12}}\)

Solution

ⓐ \({x}^{3}\) ⓑ \(|{y}^{3}|\)

Try it.

ⓐ \(\sqrt[5]{{a}^{10}}\) ⓑ \(\sqrt[3]{{b}^{27}}\)

Try it.

ⓐ \(\sqrt[4]{{m}^{8}}\) ⓑ \(\sqrt[5]{{n}^{20}}\)

Solution

ⓐ \({m}^{2}\) ⓑ \({n}^{4}\)

Try it.

ⓐ \(\sqrt[6]{{r}^{12}}\) ⓑ \(\sqrt[3]{{s}^{30}}\)

Try it.

ⓐ \(\sqrt{49{x}^{2}}\) ⓑ \(\text{-}\sqrt{81{x}^{18}}\)

Solution

ⓐ \(7|x|\) ⓑ \(-9|{x}^{9}|\)

Try it.

ⓐ \(\sqrt{100{y}^{2}}\) ⓑ \(\text{-}\sqrt{100{m}^{32}}\)

Try it.

ⓐ \(\sqrt{121{m}^{20}}\) ⓑ \(\text{-}\sqrt{64{a}^{2}}\)

Solution

ⓐ \(11{m}^{10}\) ⓑ \(-8|a|\)

Try it.

ⓐ \(\sqrt{81{x}^{36}}\) ⓑ \(\text{-}\sqrt{25{x}^{2}}\)

Try it.

ⓐ \(\sqrt[4]{16{x}^{8}}\) ⓑ \(\sqrt[6]{64{y}^{12}}\)

Solution

ⓐ \(2{x}^{2}\) ⓑ \(2{y}^{2}\)

Try it.

ⓐ \(\sqrt[3]{-8{c}^{9}}\) ⓑ \(\sqrt[3]{125{d}^{15}}\)

Try it.

ⓐ \(\sqrt[3]{216{a}^{6}}\) ⓑ \(\sqrt[5]{32{b}^{20}}\)

Solution

ⓐ \(6{a}^{2}\) ⓑ \(2{b}^{4}\)

Try it.

ⓐ \(\sqrt[7]{128{r}^{14}}\) ⓑ \(\sqrt[4]{81{s}^{24}}\)

Try it.

Solution

Try it.

Solution

Try it.

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.

Simplify: ⓐ \({(-9)}^{2}\) ⓑ \(\text{-}{9}^{2}\) ⓒ \({(-9)}^{3}.\)
If you missed this problem, review .

Jawaabta muuji

ⓐ 8; ⓑ −1; ⓒ -729
Round \(3.846\) to the nearest hundredth.
If you missed this problem, review .

Jawaabta muuji

3.85
Simplify: ⓐ \({x}^{3}\cdot {x}^{3}\) ⓑ \({y}^{2}\cdot {y}^{2}\cdot {y}^{2}\) ⓒ \({z}^{3}\cdot {z}^{3}\cdot {z}^{3}\cdot {z}^{3}.\)
If you missed this problem, review .

Jawaabta muuji

ⓐ \({x}^{6}\); ⓑ \({y}^{6}\); ⓒ \({z}^{12}\)
Simplify: ⓐ \(\sqrt{144}\) ⓑ \(\text{-}\sqrt{289}.\)

Jawaabta muuji

ⓐ

\(\sqrt{144}\)
Since \({12}^{2}=144.\) \(12\)

ⓑ

\(-\sqrt{289}\)
Since \({17}^{2}=289\) and the negative is in front of the radical sign. \(-17\)
Simplify: ⓐ \(\text{-}\sqrt{64}\) ⓑ \(\sqrt{225}.\)

Jawaabta muuji

ⓐ \(-8\) ⓑ 15
Simplify: ⓐ \(\sqrt{100}\) ⓑ \(\text{-}\sqrt{121}.\)

Jawaabta muuji

ⓐ 10 ⓑ \(-11\)
Simplify: ⓐ \(\sqrt{-196}\) ⓑ \(\text{-}\sqrt{64}.\)

Jawaabta muuji

ⓐ

\(\sqrt{-196}\)
There is no real number whose square is \(-196.\) \(\sqrt{-196}\ \text{is not a real number.}\)

ⓑ

\(-\sqrt{64}\)
The negative is in front of the radical. \(-8\)
Simplify: ⓐ \(\sqrt{-169}\) ⓑ \(\text{-}\sqrt{81}.\)

Jawaabta muuji

ⓐ not a real number ⓑ \(-9\)
Simplify: ⓐ \(\text{-}\sqrt{49}\) ⓑ \(\sqrt{-121}.\)

Jawaabta muuji

ⓐ \(-7\) ⓑ not a real number
Simplify: ⓐ \(\sqrt[3]{64}\) ⓑ \(\sqrt[4]{81}\) ⓒ \(\sqrt[5]{32}.\)

Jawaabta muuji

ⓐ

\(\sqrt[3]{64}\)
Since \({4}^{3}=64.\) \(4\)

ⓑ

\(\sqrt[4]{81}\)
Since \({(3)}^{4}=81.\) \(3\)

ⓒ

\(\sqrt[5]{32}\)
Since \({(2)}^{5}=32.\) \(2\)
Simplify: ⓐ \(\sqrt[3]{-125}\) ⓑ \(\sqrt[4]{-16}\) ⓒ \(\sqrt[5]{-243}.\)

Jawaabta muuji

ⓐ

\(\sqrt[3]{-125}\)
Since \({(-5)}^{3}=-125.\) \(-5\)

ⓑ

\(\sqrt[4]{-16}\)
Think, \({(?)}^{4}=-16.\) No real number raised to the fourth power is negative. Not a real number.

ⓒ

\(\sqrt[5]{-243}\)
Since \({(-3)}^{5}=-243.\) \(-3\)
Estimate each root between two consecutive whole numbers: ⓐ \(\sqrt{105}\) ⓑ \(\sqrt[3]{43}.\)

Jawaabta muuji

ⓐ Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.
Locate 105 between two consecutive perfect squares.
\(\sqrt{105}\) is between their square roots.
ⓑ Similarly we locate 43 between two perfect cube numbers.
Locate 43 between two consecutive perfect cubes.
\(\sqrt[3]{43}\) is between their cube roots.
Estimate each root between two consecutive whole numbers:
ⓐ \(\sqrt{38}\) ⓑ \(\sqrt[3]{93}\)

Jawaabta muuji

ⓐ \(6<\sqrt{38}<7\)
ⓑ \(4<\sqrt[3]{93}<5\)
Estimate each root between two consecutive whole numbers:
ⓐ \(\sqrt{84}\) ⓑ \(\sqrt[3]{152}\)

Jawaabta muuji

ⓐ \(9<\sqrt{84}<10\)
ⓑ \(5<\sqrt[3]{152}<6\)
Round to two decimal places: ⓐ \(\sqrt{17}\) ⓑ \(\sqrt[3]{49}\) ⓒ \(\sqrt[4]{51}.\)

Jawaabta muuji

ⓐ

\(\sqrt{17}\)
Use the calculator square root key. \(4.123105626\text{\ldots }\)
Round to two decimal places. \(4.12\)
\(\sqrt{17}\approx 4.12\)

ⓑ

\(\sqrt[3]{49}\)
Use the calculator \(\sqrt[y]{x}\) key. \(3.659305710\text{\ldots }\)
Round to two decimal places. \(3.66\)
\(\sqrt[3]{49}\approx 3.66\)

ⓒ

\(\sqrt[4]{51}\)
Use the calculator \(\sqrt[y]{x}\) key. \(2.6723451177\text{\ldots }\)
Round to two decimal places. \(2.67\)
\(\sqrt[4]{51}\approx 2.67\)
Round to two decimal places:
ⓐ \(\sqrt{11}\) ⓑ \(\sqrt[3]{71}\) ⓒ \(\sqrt[4]{127}.\)

Jawaabta muuji

ⓐ \(\approx 3.32\) ⓑ \(\approx 4.14\)
ⓒ \(\approx 3.36\)
Round to two decimal places:
ⓐ \(\sqrt{13}\) ⓑ \(\sqrt[3]{84}\) ⓒ \(\sqrt[4]{98}.\)

Jawaabta muuji

ⓐ \(\approx 3.61\) ⓑ \(\approx 4.38\)
ⓒ \(\approx 3.15\)
Simplify: ⓐ \(\sqrt{{x}^{2}}\) ⓑ \(\sqrt[3]{{n}^{3}}\) ⓒ \(\sqrt[4]{{p}^{4}}\) ⓓ \(\sqrt[5]{{y}^{5}}.\)

Jawaabta muuji

ⓐ We use the absolute value to be sure to get the positive root.
\(\sqrt{{x}^{2}}\)
Since the index \(n\) is even, \(\sqrt[n]{{a}^{n}}=|a|.\) \(|x|\)
ⓑ This is an odd indexed root so there is no need for an absolute value sign.
\(\sqrt[3]{{n}^{3}}\)
Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) \(n\)
ⓒ
\(\sqrt[4]{{p}^{4}}\)
Since the index \(n\ \text{is even}\ \sqrt[n]{{a}^{n}}=|a|.\) \(|p|\)
ⓓ
\(\sqrt[5]{{y}^{5}}\)
Since the index \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) \(y\)
Simplify: ⓐ \(\sqrt{{b}^{2}}\) ⓑ \(\sqrt[3]{{w}^{3}}\) ⓒ \(\sqrt[4]{{m}^{4}}\) ⓓ \(\sqrt[5]{{q}^{5}}.\)

Jawaabta muuji

ⓐ \(|b|\) ⓑ w ⓒ \(|m|\) ⓓ q
Simplify: ⓐ \(\sqrt{{y}^{2}}\) ⓑ \(\sqrt[3]{{p}^{3}}\) ⓒ \(\sqrt[4]{{z}^{4}}\) ⓓ \(\sqrt[5]{{q}^{5}}.\)

Jawaabta muuji

ⓐ \(|y|\) ⓑ p ⓒ \(|z|\) ⓓ q

Simplify: ⓐ \(\sqrt{{x}^{6}}\) ⓑ \(\sqrt{{y}^{16}}.\)

Jawaabta muuji

ⓐ

	\(\sqrt{{x}^{6}}\)
Since \({({x}^{3})}^{2}={x}^{6}.\)	\(\sqrt{{({x}^{3})}^{2}}\)
Since the index \(n\) is even \(\sqrt{{a}^{n}}=\|a\|.\)	\(\|{x}^{3}\|\)

ⓑ

	\(\sqrt{{y}^{16}}\)
Since \({({y}^{8})}^{2}={y}^{16}.\)	\(\sqrt{{({y}^{8})}^{2}}\)
Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=\|a\|.\)	\({y}^{8}\)
In this case the absolute value sign is not needed as \({y}^{8}\) is positive.

Simplify: ⓐ \(\sqrt{{y}^{18}}\) ⓑ \(\sqrt{{z}^{12}}.\)

Jawaabta muuji

ⓐ \(|{y}^{9}|\) ⓑ \({z}^{6}\)
Simplify: ⓐ \(\sqrt{{m}^{4}}\) ⓑ \(\sqrt{{b}^{10}}.\)

Jawaabta muuji

ⓐ \({m}^{2}\) ⓑ \(|{b}^{5}|\)
Simplify: ⓐ \(\sqrt[3]{{y}^{18}}\) ⓑ \(\sqrt[4]{{z}^{8}}.\)

Jawaabta muuji

ⓐ

\(\sqrt[3]{{y}^{18}}\)
Since \({({y}^{6})}^{3}={y}^{18}.\) \(\sqrt[3]{{({y}^{6})}^{3}}\)
Since \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\) \({y}^{6}\)

ⓑ

\(\sqrt[4]{{z}^{8}}\)
Since \({({z}^{2})}^{4}={z}^{8}.\) \(\sqrt[4]{{({z}^{2})}^{4}}\)
Since \({z}^{2}\) is positive, we do not need an absolute value sign. \({z}^{2}\)
Simplify: ⓐ \(\sqrt[4]{{u}^{12}}\) ⓑ \(\sqrt[3]{{v}^{15}}.\)

Jawaabta muuji

ⓐ \(|{u}^{3}|\) ⓑ \({v}^{5}\)
Simplify: ⓐ \(\sqrt[5]{{c}^{20}}\) ⓑ \(\sqrt[6]{{d}^{24}}\)

Jawaabta muuji

ⓐ \({c}^{4}\) ⓑ \({d}^{4}\)
Simplify: ⓐ \(\sqrt{16{n}^{2}}\) ⓑ \(\text{-}\sqrt{81{c}^{2}}.\)

Jawaabta muuji

ⓐ

\(\sqrt{16{n}^{2}}\)
Since \({(4n)}^{2}=16{n}^{2}.\) \(\sqrt{{(4n)}^{2}}\)
Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=|a|.\) \(4|n|\)

ⓑ

\(-\sqrt{81{c}^{2}}\)
Since \({(9c)}^{2}=81{c}^{2}.\) \(-\sqrt{{(9c)}^{2}}\)
Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=|a|.\) \(-9|c|\)
Simplify: ⓐ \(\sqrt{64{x}^{2}}\) ⓑ \(\text{-}\sqrt{100{p}^{2}}.\)

Jawaabta muuji

ⓐ \(8|x|\) ⓑ \(-10|p|\)
Simplify: ⓐ \(\sqrt{169{y}^{2}}\) ⓑ \(\text{-}\sqrt{121{y}^{2}}.\)

Jawaabta muuji

ⓐ \(13|y|\) ⓑ \(-11|y|\)
Simplify: ⓐ \(\sqrt[3]{64{p}^{6}}\) ⓑ \(\sqrt[4]{16{q}^{12}}.\)

Jawaabta muuji

ⓐ

\(\sqrt[3]{64{p}^{6}}\)
Rewrite \(64{p}^{6}\) as \({(4{p}^{2})}^{3}.\) \(\sqrt[3]{{(4{p}^{2})}^{3}}\)
Take the cube root. \(4{p}^{2}\)

ⓑ

\(\sqrt[4]{16{q}^{12}}\)
Rewrite the radicand as a fourth power. \(\sqrt[4]{{(2{q}^{3})}^{4}}\)
Take the fourth root. \(2|{q}^{3}|\)
Simplify: ⓐ \(\sqrt[3]{27{x}^{27}}\) ⓑ \(\sqrt[4]{81{q}^{28}}.\)

Jawaabta muuji

ⓐ \(3{x}^{9}\) ⓑ \(3|{q}^{7}|\)
Simplify: ⓐ \(\sqrt[3]{125{q}^{9}}\) ⓑ \(\sqrt[5]{243{q}^{25}}.\)

Jawaabta muuji

ⓐ \(5{q}^{3}\) ⓑ \(3{q}^{5}\)
Simplify: ⓐ \(\sqrt{36{x}^{2}{y}^{2}}\) ⓑ \(\sqrt{121{a}^{6}{b}^{8}}\) ⓒ \(\sqrt[3]{64{p}^{63}{q}^{9}}.\)

Jawaabta muuji

ⓐ

\(\sqrt{36{x}^{2}{y}^{2}}\)
Since \({(6xy)}^{2}=36{x}^{2}{y}^{2}\) \(\sqrt{{(6xy)}^{2}}\)
Take the square root. \(6|xy|\)

ⓑ

\(\sqrt{121{a}^{6}{b}^{8}}\)
Since \({(11{a}^{3}{b}^{4})}^{2}=121{a}^{6}{b}^{8}\) \(\sqrt{{(11{a}^{3}{b}^{4})}^{2}}\)
Take the square root. \(11|{a}^{3}|{b}^{4}\)

ⓒ

\(\sqrt[3]{64{p}^{63}{q}^{9}}\)
Since \({(4{p}^{21}{q}^{3})}^{3}=64{p}^{63}{q}^{9}\) \(\sqrt[3]{{(4{p}^{21}{q}^{3})}^{3}}\)
Take the cube root. \(4{p}^{21}{q}^{3}\)
Simplify: ⓐ \(\sqrt{100{a}^{2}{b}^{2}}\) ⓑ \(\sqrt{144{p}^{12}{q}^{20}}\) ⓒ \(\sqrt[3]{8{x}^{30}{y}^{12}}\)

Jawaabta muuji

ⓐ \(10|ab|\) ⓑ \(12{p}^{6}{q}^{10}\)
ⓒ \(2{x}^{10}{y}^{4}\)
Simplify: ⓐ \(\sqrt{225{m}^{2}{n}^{2}}\) ⓑ \(\sqrt{169{x}^{10}{y}^{14}}\) ⓒ \(\sqrt[3]{27{w}^{36}{z}^{15}}\)

Jawaabta muuji

ⓐ \(15|mn|\) ⓑ \(13|{x}^{5}{y}^{7}|\)
ⓒ \(3{w}^{12}{z}^{5}\)
ⓐ \(\sqrt{64}\) ⓑ \(\text{-}\sqrt{81}\)

Jawaabta muuji

ⓐ 8 ⓑ \(-9\)

Symbols used here

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

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

\approx

approximately equal
Equal to the precision shown, not exactly.

\leq,\ \geq

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

\pm

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

\neq

not equal
The two sides are different.

|x|

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

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: Simplify Expressions with Roots

Simplify expressions with roots
Estimate and approximate roots
Simplify variable expressions with roots
If
The square root of
If
The principal
When

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.

Ku day inaad ku

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.

In ka badan Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

	\(\sqrt[4]{-16}\)
Think, \({(?)}^{4}=-16.\) No real number raised to the fourth power is negative.	Not a real number.

	\(\sqrt{17}\)
Use the calculator square root key.	\(4.123105626\text{\ldots }\)
Round to two decimal places.	\(4.12\)
	\(\sqrt{17}\approx 4.12\)

	\(\sqrt[3]{49}\)
Use the calculator \(\sqrt[y]{x}\) key.	\(3.659305710\text{\ldots }\)
Round to two decimal places.	\(3.66\)
	\(\sqrt[3]{49}\approx 3.66\)

	\(\sqrt[4]{51}\)
Use the calculator \(\sqrt[y]{x}\) key.	\(2.6723451177\text{\ldots }\)
Round to two decimal places.	\(2.67\)
	\(\sqrt[4]{51}\approx 2.67\)

	\(\sqrt[3]{{y}^{18}}\)
Since \({({y}^{6})}^{3}={y}^{18}.\)	\(\sqrt[3]{{({y}^{6})}^{3}}\)
Since \(n\) is odd, \(\sqrt[n]{{a}^{n}}=a.\)	\({y}^{6}\)

	\(\sqrt[4]{{z}^{8}}\)
Since \({({z}^{2})}^{4}={z}^{8}.\)	\(\sqrt[4]{{({z}^{2})}^{4}}\)
Since \({z}^{2}\) is positive, we do not need an absolute value sign.	\({z}^{2}\)

	\(\sqrt{16{n}^{2}}\)
Since \({(4n)}^{2}=16{n}^{2}.\)	\(\sqrt{{(4n)}^{2}}\)
Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=\|a\|.\)	\(4\|n\|\)

	\(-\sqrt{81{c}^{2}}\)
Since \({(9c)}^{2}=81{c}^{2}.\)	\(-\sqrt{{(9c)}^{2}}\)
Since the index \(n\) is even \(\sqrt[n]{{a}^{n}}=\|a\|.\)	\(-9\|c\|\)

	\(\sqrt[3]{64{p}^{6}}\)
Rewrite \(64{p}^{6}\) as \({(4{p}^{2})}^{3}.\)	\(\sqrt[3]{{(4{p}^{2})}^{3}}\)
Take the cube root.	\(4{p}^{2}\)

	\(\sqrt[4]{16{q}^{12}}\)
Rewrite the radicand as a fourth power.	\(\sqrt[4]{{(2{q}^{3})}^{4}}\)
Take the fourth root.	\(2\|{q}^{3}\|\)

	\(\sqrt[3]{64}\)
Since \({4}^{3}=64.\)	\(4\)

	\(\sqrt[4]{81}\)
Since \({(3)}^{4}=81.\)	\(3\)

	\(\sqrt[5]{32}\)
Since \({(2)}^{5}=32.\)	\(2\)

	\(\sqrt[5]{-243}\)
Since \({(-3)}^{5}=-243.\)	\(-3\)

	\(\sqrt{36{x}^{2}{y}^{2}}\)
Since \({(6xy)}^{2}=36{x}^{2}{y}^{2}\)	\(\sqrt{{(6xy)}^{2}}\)
Take the square root.	\(6\|xy\|\)

	\(\sqrt{121{a}^{6}{b}^{8}}\)
Since \({(11{a}^{3}{b}^{4})}^{2}=121{a}^{6}{b}^{8}\)	\(\sqrt{{(11{a}^{3}{b}^{4})}^{2}}\)
Take the square root.	\(11\|{a}^{3}\|{b}^{4}\)

	\(\sqrt[3]{64{p}^{63}{q}^{9}}\)
Since \({(4{p}^{21}{q}^{3})}^{3}=64{p}^{63}{q}^{9}\)	\(\sqrt[3]{{(4{p}^{21}{q}^{3})}^{3}}\)
Take the cube root.	\(4{p}^{21}{q}^{3}\)