Simplify and Use 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 “$n$ squared.” For example, $8^2$ is read as “$8$ squared.” We call $64$ the square of $8$ because $8^2 = 64$. Similarly, $121$ is the square of $11$, because $11^2 = 121$.

Modeling squares

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:

The number $9$ is called a perfect square because it is the square of a whole number. Here are the squares of the counting numbers $1$ through $15$:

What happens when you square a negative number?

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive. In fact, the squares of the negative integers from $-1$ to $-15$ are exactly the same as the squares of the positive integers from $1$ to $15$ shown in the table above.

Square roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because $10^2 = 100$, we say $100$ is the square of $10$. We can also say that $10$ is a square root of $100$.

Notice $(-10)^2 = 100$ also, so $-10$ is also a square root of $100$. Therefore, both $10$ and $-10$ are square roots of $100$. So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, $\sqrt{\phantom{x}}$, stands for the positive square root. The positive square root is also called the principal square root.

We can 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. The chart below shows the square roots of the first $15$ perfect square numbers:

Example. Simplify: (a) $\sqrt{25}$ (b) $\sqrt{121}$.

(a) Since $5^2 = 25$: $\sqrt{25} = 5$.

(b) Since $11^2 = 121$: $\sqrt{121} = 11$.

Every positive number has two square roots, and the radical sign indicates the positive one. We write $\sqrt{100} = 10$. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, $-\sqrt{100} = -10$.

Example. Simplify: (a) $-\sqrt{9}$ (b) $-\sqrt{144}$.

(a) The negative is in front of the radical sign: $-\sqrt{9} = -3$.

(b) The negative is in front of the radical sign: $-\sqrt{144} = -12$.

Square root of a negative number

Can we simplify $\sqrt{-25}$? Is there a number whose square is $-25$? None of the numbers that we have dealt with so far have a square that is $-25$. Why? Any positive number squared is positive, and any negative number squared is also positive. So we say there is no real number equal to $\sqrt{-25}$. If we are asked to find the square root of any negative number, we say that the solution is not a real number.

Example. Simplify: (a) $\sqrt{-169}$ (b) $-\sqrt{121}$.

(a) There is no real number whose square is $-169$. Therefore, $\sqrt{-169}$ is not a real number.

(b) The negative is in front of the radical sign, so we find the opposite of the square root of $121$: $-\sqrt{121} = -11$.

Square roots and the order of operations

When using the order of operations to simplify an expression that has square roots, we treat the radical sign as a grouping symbol. We simplify any expressions under the radical sign before performing other operations.

Example. Simplify: (a) $\sqrt{25} + \sqrt{144}$ (b) $\sqrt{25 + 144}$.

(a) Simplify each radical first, then add: $5 + 12 = 17$.

(b) Add under the radical sign first, then simplify: $\sqrt{169} = 13$.

Notice the different answers in (a) and (b) — it is important to follow the order of operations correctly.

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 this pattern, we could say that $\sqrt{5}$ is between $2$ and $3$. Using inequality symbols, we write $2 < \sqrt{5} < 3$.

Example. Estimate $\sqrt{60}$ between two consecutive whole numbers.

Think of the perfect squares closest to $60$: $49 < 60 < 64$. Since $60$ is between the perfect squares $49$ and $64$, $\sqrt{60}$ 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. 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 $10$-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: $2.236067978^2 = 5.000000002$ and $2.24^2 = 5.0176$. The squares are close, but not exactly equal, to $5$.

Example. Round $\sqrt{17}$ to two decimal places using a calculator.

Use the calculator square root key: $4.123105626$. Round to two decimal places: $\sqrt{17} \approx 4.12$.

Simplify variable expressions with square roots

Expressions with square roots 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 $\sqrt{9x^2}$, where $x \geq 0$. Can you think of an expression whose square is $9x^2$? $(3x)^2 = 9x^2$, so $\sqrt{9x^2} = 3x$.

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. Simplify: $\sqrt{x^2}$.

Think about what we would have to square to get $x^2$: since $(x)^2 = x^2$, $\sqrt{x^2} = x$.

Example. Simplify: $\sqrt{16x^2}$.

Since $(4x)^2 = 16x^2$: $\sqrt{16x^2} = 4x$.

Example. Simplify: $-\sqrt{81y^2}$.

Since $(9y)^2 = 81y^2$: $-\sqrt{81y^2} = -9y$.

Example. Simplify: $\sqrt{36x^2y^2}$.

Since $(6xy)^2 = 36x^2y^2$: $\sqrt{36x^2y^2} = 6xy$.

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.

Square roots and area

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. For example, if the area is $9$ square units, the side is $\sqrt{9} = 3$ units; if the area is $144$ square units, the side is $\sqrt{144} = 12$ units.

Example. 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?

We know the area of the square is $200$ square feet and want to find the length of the side. Translate to an expression: $\sqrt{A}$. Evaluate $\sqrt{A}$ when $A = 200$: $\sqrt{200}$. Use your calculator: $14.142135\ldots$. Round to one decimal place: $14.1$ feet. Each side of the patio should be $14.1$ feet.

Square roots and gravity

Another application of square roots involves gravity. On Earth, if an object is dropped from a height of $h$ feet, the time in seconds it will take to reach the ground is found by evaluating the expression $\tfrac{\sqrt{h}}{4}$. For example, if an object is dropped from a height of $64$ feet, we can find the time it takes to reach the ground by evaluating $\tfrac{\sqrt{64}}{4} = \tfrac{8}{4} = 2$. It would take $2$ seconds for an object dropped from a height of $64$ feet to reach the ground.

Example. 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?

Translate to an expression: $\tfrac{\sqrt{h}}{4}$. Evaluate $\tfrac{\sqrt{h}}{4}$ when $h = 400$: $\tfrac{\sqrt{400}}{4}$. Find the square root of $400$: $\tfrac{20}{4}$. Simplify: $5$. It will take $5$ seconds for the sunglasses to reach the river.

Square roots and accident investigations

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes. According to some formulas, if the length of the skid marks is $d$ feet, then the speed of the car can be found by evaluating $\sqrt{24d}$.

Example. 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?

Translate to an expression: $\sqrt{24d}$. Evaluate $\sqrt{24d}$ when $d = 190$: $\sqrt{24 \cdot 190} = \sqrt{4{,}560}$. Use your calculator: $67.527772\ldots$. Round to tenths: $67.5$. The speed of the car was approximately $67.5$ miles per hour.

Key terms

perfect square — the square of a whole number. square root — a number whose square is $m$; if $n^2 = m$, then $n$ is a square root of $m$. radical sign — the symbol $\sqrt{\phantom{x}}$, which denotes the principal (positive) square root. radicand — the number or expression under a radical sign. principal square root — the positive square root of a positive number.

This section is adapted from Prealgebra 2e, Section 5.7: Simplify and Use Square Roots by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated the perfect-square and square-root charts, the area/side-length table, and the tile-diagram concept as markdown tables and prose; omitted the Be Prepared quiz, Manipulative Mathematics and Links to Literacy callouts, Media links, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.