Lesson 8.1: Simplify Expressions with Roots

New Concept

This lesson introduces roots as the inverse of exponents. You'll learn to simplify numerical and variable expressions like $\sqrt[n]{a}$, estimate values, and see why for even roots, $\sqrt[n]{a^n} = |a|$, ensuring a positive principal root.

What’s next

Now, let's build on this foundation. You'll work through interactive examples and practice cards to master simplifying various root expressions and variables.

Property

If $n^2 = m$, then $m$ is the square of $n$.
If $n^2 = m$, then $n$ is a square root of $m$.
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Square Root Notation
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$\sqrt{m}$ is read "the square root of $m$".
If $n^2 = m$, then $n = \sqrt{m}$, for $n \geq 0$.
The symbol $\sqrt{\phantom{m}}$ is called a radical sign. The expression under the radical sign is called the radicand. The positive square root is also called the principal square root.

Examples

• To simplify $\sqrt{36}$, we find the number that when squared gives 36. Since $6^2 = 36$, $\sqrt{36} = 6$.

• To simplify $-\sqrt{49}$, we first find $\sqrt{49} = 7$. The negative sign is outside the radical, so the final answer is $-7$.

Property

If $b^n = a$, then $b$ is an $n^{\text{th}}$ root of $a$.
The principal $n^{\text{th}}$ root of $a$ is written $\sqrt[n]{a}$.
$n$ is called the index of the radical.
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Properties of $\sqrt[n]{a}$
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When $n$ is an even number and:

• $a \geq 0$, 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 $a$.

Examples

• To simplify $\sqrt[3]{125}$, we look for a number that, when cubed, is 125. Since $5^3 = 125$, $\sqrt[3]{125} = 5$.

Property

To get a numerical estimate of a root, we look for perfect powers closest to the radicand. For example, to estimate $\sqrt{11}$, we see 11 is between the perfect squares 9 and 16. Its square root will be between 3 and 4. For a decimal approximation, a calculator is used. The symbol for an approximation is $\approx$.

Examples

• To estimate $\sqrt{50}$, we know $7^2 = 49$ and $8^2 = 64$. Since $49 < 50 < 64$, we can state that $7 < \sqrt{50} < 8$.

• To estimate $\sqrt[3]{70}$, we know $4^3 = 64$ and $5^3 = 125$. Since $64 < 70 < 125$, we can state that $4 < \sqrt[3]{70} < 5$.

Property

For any integer $n \geq 2$,

• when the index $n$ is odd, $\sqrt[n]{a^n} = a$
• when the index $n$ 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.

Examples

• Simplify $\sqrt{y^2}$. Since the index (2) is even, we must use an absolute value. The result is $|y|$.

Property

To simplify roots of variables with higher powers, use the Power Property of Exponents, $(a^m)^n = a^{mn}$, in reverse. For a square root, $\sqrt{a^{2m}} = \sqrt{(a^m)^2} = |a^m|$. In general, $\sqrt[n]{x^{kn}} = \sqrt[n]{(x^k)^n}$, which simplifies to $|x^k|$ if $n$ is even, and $x^k$ if $n$ is odd.

Examples

• Simplify $\sqrt{x^{10}}$. This is $\sqrt{(x^5)^2}$, which is $|x^5|$. The absolute value is needed because the index (2) is even and the new exponent (5) is odd.

• Simplify $\sqrt[3]{y^{15}}$. This is $\sqrt[3]{(y^5)^3}$, which is $y^5$. No absolute value is needed because the index (3) is odd.

Property

The concept $\sqrt{a^{2m}} = |a^m|$ extends to expressions with coefficients. To simplify, handle the coefficient and the variable parts separately. For example, to simplify $\sqrt{16r^{22}}$, you find $\sqrt{16}$ and $\sqrt{r^{22}}$ separately and combine them: $4|r^{11}|$.

Examples

• Simplify $\sqrt{49x^2}$. This is $\sqrt{49} \cdot \sqrt{x^2} = 7|x|$.

• Simplify $\sqrt[3]{27y^9}$. This is $\sqrt[3]{27} \cdot \sqrt[3]{y^9} = 3y^3$.