Lesson 9.7: Higher Roots

New Concept

This lesson expands on square roots by introducing higher roots, like cube roots ($\sqrt[3]{a}$) and fourth roots ($\sqrt[4]{a}$). You'll learn to simplify, add, and subtract these expressions by applying the product and quotient properties for radicals.

What’s next

Next, you'll dive into interactive examples and practice problems, applying the Product and Quotient Properties to simplify and combine higher roots.

Property

If $b^n = a$, then $b$ is an nth root of a number $a$. The principal nth 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 $a$.

Examples

• To simplify $\sqrt[3]{125}$, we ask 'what number cubed is 125?'. Since $5^3 = 125$, $\sqrt[3]{125} = 5$.
• To simplify $\sqrt[4]{-81}$, we see the index (4) is even and the number inside is negative. Therefore, it is not a real number.
• To simplify $\sqrt[5]{-32}$, we ask 'what number to the fifth power is -32?'. Since $(-2)^5 = -32$, $\sqrt[5]{-32} = -2$.

Property

For any integer $n \ge 2$,
when $n$ is odd $\sqrt[n]{a^n} = a$
when $n$ is even $\sqrt[n]{a^n} = |a|$

Examples

• To simplify $\sqrt[6]{x^6}$, the index is even, so we must use an absolute value. The result is $|x|$.
• To simplify $\sqrt[7]{m^7}$, the index is odd, so we do not use an absolute value. The result is $m$.
• To simplify $\sqrt[4]{(3y)^4}$, the index is even. The result is $|3y|$, which simplifies to $3|y|$ since 3 is positive.

Explanation

When taking an even root of a variable expression, use absolute value signs. This ensures the answer is positive, as principal even roots must be non-negative. For odd roots, the sign is preserved, so no absolute value is needed.

Property

$\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ and $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$
when $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers and for any integer $n \ge 2$.
An nth root is considered simplified if it has no factors of $m^n$.

Examples

• To simplify $\sqrt[3]{40}$, we find the largest perfect cube factor, which is 8. So, $\sqrt[3]{8 \cdot 5} = \sqrt[3]{8} \cdot \sqrt[3]{5} = 2\sqrt[3]{5}$.
• To simplify $\sqrt[4]{32x^5}$, we look for perfect fourth powers: $\sqrt[4]{16x^4 \cdot 2x} = \sqrt[4]{(2x)^4} \cdot \sqrt[4]{2x} = 2|x|\sqrt[4]{2x}$.
• To simplify $\sqrt[3]{-54y^8}$, we find perfect cube factors: $\sqrt[3]{-27y^6 \cdot 2y^2} = \sqrt[3]{(-3y^2)^3} \cdot \sqrt[3]{2y^2} = -3y^2\sqrt[3]{2y^2}$.

Explanation

This property lets you split a radical into the product of smaller radicals. To simplify, find the largest perfect nth power factor inside the radical, take its root, and leave the rest of the factors inside.

Property

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b \ne 0$, and for any integer $n \ge 2$

Examples

• To simplify $\frac{\sqrt[4]{162y^5}}{\sqrt[4]{2y}}$, we combine them: $\sqrt[4]{\frac{162y^5}{2y}} = \sqrt[4]{81y^4} = 3|y|$.
• To simplify $\sqrt[3]{\frac{x^9}{x^3}}$, first simplify the fraction inside: $\sqrt[3]{x^6} = \sqrt[3]{(x^2)^3} = x^2$.
• To simplify $\sqrt[3]{\frac{16a^5}{b^6}}$, we use the property to split it: $\frac{\sqrt[3]{16a^5}}{\sqrt[3]{b^6}} = \frac{\sqrt[3]{8a^3 \cdot 2a^2}}{\sqrt[3]{(b^2)^3}} = \frac{2a\sqrt[3]{2a^2}}{b^2}$.

Explanation

This property allows you to rewrite a radical of a fraction as a fraction of radicals. You can also use it in reverse to combine two radicals that are in a fraction into one single radical.

Property

Radicals with the same index and same radicand are called like radicals.
We add and subtract like radicals in the same way we add and subtract like terms.

Examples

• To simplify $5\sqrt[4]{10z} - 3\sqrt[4]{10z}$, the radicals are like. We subtract the coefficients: $(5-3)\sqrt[4]{10z} = 2\sqrt[4]{10z}$.
• To simplify $\sqrt[3]{16} + \sqrt[3]{128}$, we first simplify each radical: $\sqrt[3]{8 \cdot 2} + \sqrt[3]{64 \cdot 2} = 2\sqrt[3]{2} + 4\sqrt[3]{2}$. Now they are like radicals, so we add: $6\sqrt[3]{2}$.
• To simplify $\sqrt[3]{40x^5} - \sqrt[3]{5x^2}$, we simplify the first term: $\sqrt[3]{8x^3 \cdot 5x^2} - \sqrt[3]{5x^2} = 2x\sqrt[3]{5x^2} - 1\sqrt[3]{5x^2} = (2x-1)\sqrt[3]{5x^2}$.

Explanation

Think of like radicals as like terms. If the index and the number inside the radical (radicand) are identical, you can add or subtract their coefficients. If they are not identical, try simplifying them first!