Lesson 9.1: Simplify and Use Square Roots

New Concept

Discover the concept of a square root, the inverse of squaring a number. This lesson will teach you how to simplify, estimate, and approximate square roots for both numbers and variable expressions.

What’s next

Next, you'll dive into worked examples and interactive practice cards to master simplifying, estimating, and approximating square roots.

Property

Square of a Number
If $n^2 = m$, then $m$ is the square of $n$.

Square Root of a Number
If $n^2 = m$, then $n$ is a square root of $m$.

Square Root Notation
$\sqrt{m}$ is read as “the square root of $m$.”
If $m = n^2$, then $\sqrt{m} = n$, for $n \ge 0$.
The square root of $m$, $\sqrt{m}$, is the positive number whose square is $m$. This is also called the principal square root.
To find the negative square root of a number, we place a negative in front of the radical sign.
When using the order of operations, we treat the radical as a grouping symbol.

Property

To estimate a square root between two consecutive whole numbers, first locate the number between two consecutive perfect squares.
Its square root will then lie between the square roots of those perfect squares.

Examples

• To estimate $\sqrt{20}$, we know that 16 and 25 are the closest perfect squares. Since $16 < 20 < 25$, we can say that $4 < \sqrt{20} < 5$.
• To estimate $\sqrt{90}$, we look for perfect squares near 90. Since $81 < 90 < 100$, the estimate is $9 < \sqrt{90} < 10$.
• To estimate $\sqrt{150}$, we see that $144 < 150 < 169$. Therefore, we know that $12 < \sqrt{150} < 13$.

Explanation

This method works because as numbers get bigger, their square roots also get bigger. By finding the two perfect square neighbors for your number, you can trap its square root between two whole numbers, giving you a great estimate.

Property

For numbers that are not perfect squares, the square root is an irrational number.
We use a calculator to find a decimal approximation. The symbol $\approx$ means 'is approximately equal to.'

Examples

• To approximate $\sqrt{23}$ rounded to two decimal places, a calculator shows $4.7958...$. We round this to $4.80$. So, $\sqrt{23} \approx 4.80$.
• To approximate $\sqrt{51}$ rounded to two decimal places, a calculator shows $7.1414...$. We round this to $7.14$. So, $\sqrt{51} \approx 7.14$.
• To approximate $\sqrt{2}$ rounded to two decimal places, a calculator shows $1.4142...$. We round this to $1.41$. So, $\sqrt{2} \approx 1.41$.

Explanation

The square root of a non-perfect square has a decimal that never ends or repeats. A calculator gives us a practical, rounded value to work with. This approximation is close enough for most real-world problems.

Property

To find the square root of a variable expression, use the Power Property of Exponents in reverse.
Since $(a^m)^2 = a^{2m}$, it follows that $\sqrt{a^{2m}} = a^m$.
We assume all variables represent non-negative numbers.
To simplify, take the square root of the coefficient and divide the exponent of each variable by 2.

Examples

• To simplify $\sqrt{z^{14}}$, we divide the exponent by 2. So, $\sqrt{z^{14}} = z^{14/2} = z^7$.
• To simplify $\sqrt{25x^{18}}$, we find $\sqrt{25}=5$ and divide the exponent of $x$ by 2. So, $\sqrt{25x^{18}} = 5x^9$.
• To simplify $\sqrt{144m^{12}n^6}$, we find $\sqrt{144}=12$ and divide each exponent by 2. So, $\sqrt{144m^{12}n^6} = 12m^6n^3$.

Explanation

Finding the square root 'undoes' squaring. When you square a power, you multiply the exponent by 2. So, to take the square root of a variable with an even power, you do the opposite: divide the exponent by 2.