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Simplify and Use Square Roots
Simplify expressions with square roots
Simplify Expressions with Square Roots
To start this section, we need to review some important vocabulary and notation.
Remember that when a number \(n\) is multiplied by itself, we can write this as \({n}^{2},\) which we read aloud as \(\text{“}\text{n}\ \text{squared.”}\) For example, \({8}^{2}\) is read as \(\text{“8}\ \text{squared.”}\)
We call \(64\) the square of \(8\) because \({8}^{2}=64.\) Similarly, \(121\) is the square of \(11,\) because \({11}^{2}=121.\)
Do you know why we use the word square? If we construct a square with three tiles on each side, the total number of tiles would be nine.
This is why we say that the square of three is nine.
\[{3}^{2}=9\]The number \(9\) is called a perfect square because it is the square of a whole number.
The chart shows the squares of the counting numbers \(1\) through \(15.\) You can refer to it to help you identify the perfect squares.
What happens when you square a negative number?
\[\begin{array}{ll}{(-8)}^{2} & =(-8)(-8) \\ & =64\end{array}\]When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.
The chart shows the squares of the negative integers from \(-1\) to \(-15.\)
Did you notice that these squares are the same as the squares of the positive numbers?
Condensed — the full section is in OpenStax Prealgebra 2e.
Estimate Square Roots
So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.
We might conclude that the square roots of numbers between \(4\) and \(9\) will be between \(2\) and \(3,\) and they will not be whole numbers. Based on the pattern in the table above, we could say that \(\sqrt{5}\) is between \(2\) and \(3.\) Using inequality symbols, we write
\[2<\sqrt{5}<3\]Example
Try it.
Estimate \(\sqrt{60}\) between two consecutive whole numbers.
Solution
Think of the perfect squares closest to \(60.\) Make a small table of these perfect squares and their squares roots.
| \(\text{Locate 60 between two consecutive perfect squares.}\) | \(49<60<64\) |
| \(\sqrt{60}\ \text{is between their square roots.}\) | \(7<\sqrt{60}<8\) |
Approximate Square Roots with a Calculator
There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the \(\sqrt{0}\) or \(\sqrt{x}\) key on your calculator. You will need to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, 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 \(\text{10-digit}\) display. Using it to find the square root of \(5\) will give \(2.236067977.\) This is the approximate square root of \(5.\) When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.
\[\sqrt{5}\approx 2.236067978\]You will seldom use this many digits for applications in algebra. So, if you wanted to round \(\sqrt{5}\) to two decimal places, you would write
\[\sqrt{5}\approx 2.24\]How do we know these values are approximations and not the exact values? Look at what happens when we square them.
\[\begin{array}{lll}{2.236067978}^{2} & = & 5.000000002 \\ {2.24}^{2} & = & 5.0176\end{array}\]The squares are close, but not exactly equal, to \(5.\)
Example
Try it.
Round \(\sqrt{17}\) to two decimal places using a calculator.
Solution
| \(\sqrt{17}\) | |
| Use the calculator square root key. | \(4.123105626\) |
| Round to two decimal places. | \(4.12\) |
| \(\sqrt{17}\approx 4.12\) |
Simplify Variable Expressions with Square Roots
Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?
Consider \(\sqrt{9{x}^{2}},\) where \(x\ge 0.\) Can you think of an expression whose square is \(9{x}^{2}?\)
\[\begin{array}{lll}{(?)}^{2} & = & 9{x}^{2} \\ {(3x)}^{2} & = & 9{x}^{2}\ \text{so}\ \sqrt{9{x}^{2}}=3x\end{array}\]When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.
Example
Try it.
Simplify: \(\sqrt{{x}^{2}}.\)
Solution
Think about what we would have to square to get \({x}^{2}\). Algebraically, \({(?)}^{2}={x}^{2}\)
| \(\sqrt{{x}^{2}}\) | |
| Since \({(x)}^{2}={x}^{2}\) | \(x\) |
Example
Try it.
Simplify: \(\sqrt{16{x}^{2}}.\)
Solution
| \(\sqrt{16{x}^{2}}\) | |
| \(\text{Since}\ {(4x)}^{2}=16{x}^{2}\) | \(4x\) |
Example
Try it.
Simplify: \(-\sqrt{81{y}^{2}}.\)
Solution
| \(-\sqrt{81{y}^{2}}\) | |
| \(\text{Since}\ {(9y)}^{2}=81{y}^{2}\) | \(-9y\) |
Example
Try it.
Simplify: \(\sqrt{36{x}^{2}{y}^{2}}.\)
Solution
| \(\sqrt{36{x}^{2}{y}^{2}}\) | |
| \(\text{Since}\ {(6xy)}^{2}=36{x}^{2}{y}^{2}\) | \(6xy\) |
Use Square Roots in Applications
As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!
We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.
If the area of the square is \(A\) square units, the length of a side is \(\sqrt{A}\) units. See .
| Area (square units) | Length of side (units) |
| \(9\) | \(\sqrt{9}=3\) |
| \(144\) | \(\sqrt{144}=12\) |
| \(A\) | \(\sqrt{A}\) |
Example
Try it.
Mike and Lychelle want to make a square patio. They have enough concrete for an area of \(200\) square feet. To the nearest tenth of a foot, how long can a side of their square patio be?
Solution
We know the area of the square is \(200\) square feet and want to find the length of the side. If the area of the square is \(A\) square units, the length of a side is \(\sqrt{A}\) units.
| What are you asked to find? | The length of each side of a square patio |
| Write a phrase. | The length of a side |
| Translate to an expression. | \(\sqrt{A}\) |
| Evaluate \(\sqrt{A}\) when \(A=200\). | \(\sqrt{200}\) |
| Use your calculator. | \(14.142135...\) |
| Round to one decimal place. | \(\text{14.1 feet}\) |
| Write a sentence. | Each side of the patio should be \(14.1\) feet. |
Condensed — the full section is in OpenStax Prealgebra 2e.
Key Concepts
- Square Root Notation \(\sqrt{m}\) is read ‘the square root of \(m\)’
If \(m={n}^{2}\), then \(\sqrt{m}=n\), for \(n\ge 0\). - Use a strategy for applications with square roots.
- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
- Simplify the expression.
- Write a complete sentence that answers the question.
Chapter Practice Test
In the following exercises, simplify each expression.
In the following exercises, solve.
In the following exercises, simplify.
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: \({(-9)}^{2}.\)
If you missed this problem, review .ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(81\)
-
Round \(3.846\) to the nearest hundredth.
If you missed this problem, review .ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(3.85\)
-
Evaluate \(12d\) for \(d=80.\)
If you missed this problem, review .ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(960\)
-
Simplify: ⓐ \(\ \sqrt{25}\\)ⓑ \(\ \sqrt{121}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(\sqrt{25}\) Since \({5}^{2}=25\) \(5\) ⓑ \(\sqrt{121}\) Since \({11}^{2}=121\) \(11\) -
Simplify: ⓐ \(\ \sqrt{36}\\)ⓑ \(\ \sqrt{169}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ 6
- ⓑ 13
-
Simplify: ⓐ \(\ \sqrt{16}\\)ⓑ \(\ \sqrt{196}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ 4
- ⓑ 14
-
Simplify. ⓐ \(\ -\sqrt{9}\\)ⓑ \(\ -\sqrt{144.}\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(-\sqrt{9}\) The negative is in front of the radical sign. \(-3\) ⓑ \(-\sqrt{144}\) The negative is in front of the radical sign. \(-12\) -
Simplify: ⓐ \(\ -\sqrt{4}\\)ⓑ \(\ -\sqrt{225}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ −2
- ⓑ −15
-
Simplify: ⓐ \(\ -\sqrt{81}\\)ⓑ \(\ -\sqrt{64}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ −9
- ⓑ −8
-
Simplify: ⓐ \(\ \sqrt{-169}\\)ⓑ \(\ -\sqrt{121}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ There is no real number whose square is \(-169.\) Therefore, \(\sqrt{-169}\) is not a real number.
ⓑ The negative is in front of the radical sign, so we find the opposite of the square root of \(121.\)
\(-\sqrt{121}\) The negative is in front of the radical. \(-11\) -
Simplify: ⓐ \(\ \sqrt{-196}\\)ⓑ \(\ -\sqrt{81}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ not a real number
- ⓑ −9
-
Simplify: ⓐ \(\ \sqrt{-49}\\)ⓑ \(\ -\sqrt{121}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ not a real number
- ⓑ −11
-
Simplify: ⓐ \(\ \sqrt{25}+\sqrt{144}\\)ⓑ \(\ \sqrt{25+144}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ Use the order of operations. \(\sqrt{25}+\sqrt{144}\) Simplify each radical. \(5+12\) Add. \(17\) ⓑ Use the order of operations. \(\sqrt{25+144}\) Add under the radical sign. \(\sqrt{169}\) Simplify. \(13\) -
Simplify: ⓐ \(\ \sqrt{9}+\sqrt{16}\\)ⓑ \(\ \sqrt{9+16}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ 7
- ⓑ 5
-
Simplify: ⓐ \(\ \sqrt{64+225}\\)ⓑ \(\ \sqrt{64}+\sqrt{225}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ 17
- ⓑ 23
-
Estimate \(\sqrt{60}\) between two consecutive whole numbers.
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Think of the perfect squares closest to \(60.\) Make a small table of these perfect squares and their squares roots.
\(\text{Locate 60 between two consecutive perfect squares.}\) \(49<60<64\) \(\sqrt{60}\ \text{is between their square roots.}\) \(7<\sqrt{60}<8\) -
Estimate \(\sqrt{38}\) between two consecutive whole numbers.
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(6<\sqrt{38}<7\)
-
Estimate \(\sqrt{84}\) between two consecutive whole numbers.
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(9<\sqrt{84}<10\)
-
Round \(\sqrt{17}\) to two decimal places using a calculator.
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\sqrt{17}\) Use the calculator square root key. \(4.123105626\) Round to two decimal places. \(4.12\) \(\sqrt{17}\approx 4.12\) -
Round \(\sqrt{11}\) to two decimal places.
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
≈ 3.32
-
Round \(\sqrt{13}\) to two decimal places.
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
≈ 3.61
-
Simplify: \(\sqrt{{x}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
Think about what we would have to square to get \({x}^{2}\). Algebraically, \({(?)}^{2}={x}^{2}\)
\(\sqrt{{x}^{2}}\) Since \({(x)}^{2}={x}^{2}\) \(x\) -
Simplify: \(\sqrt{{y}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
y
-
Simplify: \(\sqrt{{m}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
m
-
Simplify: \(\sqrt{16{x}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\sqrt{16{x}^{2}}\) \(\text{Since}\ {(4x)}^{2}=16{x}^{2}\) \(4x\) -
Simplify: \(\sqrt{64{x}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
8x
-
Simplify: \(\sqrt{169{y}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
13y
-
Simplify: \(-\sqrt{81{y}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(-\sqrt{81{y}^{2}}\) \(\text{Since}\ {(9y)}^{2}=81{y}^{2}\) \(-9y\) -
Simplify: \(-\sqrt{121{y}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
−11y
-
Simplify: \(-\sqrt{100{p}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
−10p
-
Simplify: \(\sqrt{36{x}^{2}{y}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\sqrt{36{x}^{2}{y}^{2}}\) \(\text{Since}\ {(6xy)}^{2}=36{x}^{2}{y}^{2}\) \(6xy\) -
Simplify: \(\sqrt{100{a}^{2}{b}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
10ab
-
Simplify: \(\sqrt{225{m}^{2}{n}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
15mn
-
Mike and Lychelle want to make a square patio. They have enough concrete for an area of \(200\) square feet. To the nearest tenth of a foot, how long can a side of their square patio be?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
We know the area of the square is \(200\) square feet and want to find the length of the side. If the area of the square is \(A\) square units, the length of a side is \(\sqrt{A}\) units.
What are you asked to find? The length of each side of a square patio Write a phrase. The length of a side Translate to an expression. \(\sqrt{A}\) Evaluate \(\sqrt{A}\) when \(A=200\). \(\sqrt{200}\) Use your calculator. \(14.142135...\) Round to one decimal place. \(\text{14.1 feet}\) Write a sentence. Each side of the patio should be \(14.1\) feet. -
Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of \(370\) square feet. To the nearest tenth of a foot, how long can a side of her square lawn be?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
19.2 feet
-
Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of \(2704\) square centimeters. How long can a side of his mosaic be?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
52 centimeters
-
Christy dropped her sunglasses from a bridge \(400\) feet above a river. How many seconds does it take for the sunglasses to reach the river?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
What are you asked to find? The number of seconds it takes for the sunglasses to reach the river Write a phrase. The time it will take to reach the river Translate to an expression. \(\frac{\sqrt{h}}{4}\) Evaluate \(\frac{\sqrt{h}}{4}\) when \(h=400\). \(\frac{\sqrt{400}}{4}\) Find the square root of 400. \(\frac{20}{4}\) Simplify. \(5\) Write a sentence. It will take 5 seconds for the sunglasses to reach the river. -
A helicopter drops a rescue package from a height of \(1296\) feet. How many seconds does it take for the package to reach the ground?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
9 seconds
-
A window washer drops a squeegee from a platform \(196\) feet above the sidewalk. How many seconds does it take for the squeegee to reach the sidewalk?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
3.5 seconds
-
After a car accident, the skid marks for one car measured \(190\) feet. To the nearest tenth, what was the speed of the car (in mph) before the brakes were applied?
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
What are you asked to find? The speed of the car before the brakes were applied Write a phrase. The speed of the car Translate to an expression. \(\sqrt{24d}\) Evaluate\(\ \sqrt{24d}\\)when\(\ d=190.\) \(\sqrt{24\cdot 190}\) Multiply. \(\sqrt{4,560}\) Use your calculator. \(67.527772...\) Round to tenths. \(67.5\) Write a sentence. The speed of the car was approximately 67.5 miles per hour.
Symbols used here
The non-negative number whose square (n-th power) is x.
Equal to the precision shown, not exactly.
Inequalities that allow equality; < and > exclude it.
Both signs at once: x = 3 ± 2 means 5 and 1.
The two sides are different.
What is left after dividing a by n.
Per hundred: 15% = 15/100.
a for every b; a divided by b.
How to: Simplify and Use Square Roots
- Simplify expressions with square roots
- Estimate square roots
- Approximate square roots
- Simplify variable expressions with square roots
- Use square roots in applications
- Identify what you are asked to find.
- Write a phrase that gives the information to find it.
- Translate the phrase to an expression.
Questions people ask
Why does the order of operations matter?
Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.
How do I check an arithmetic answer?
Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.
Why are fractions harder than decimals?
They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.
ನಿಮ್ಮದೇ ಆದದ್ದನ್ನು ಪ್ರಯತ್ನಿಸಿ
Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.