Lesson 8.2: Simplify Radical Expressions

New Concept

A radical expression $\sqrt[n]{a}$ is simplified if its radicand $a$ has no factors of $m^n$. We use the Product Property ($\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$) and Quotient Property ($\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$) to simplify.

What’s next

Now, let's break down these properties with interactive examples and practice cards to build your skills.

Property

For real numbers $a$ and $m$, and $n \geq 2$, $\sqrt[n]{a}$ is considered simplified if it has no factors of $m^n$. So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index. For example, $\sqrt{5}$ is considered simplified because there are no perfect square factors in 5. But $\sqrt{12}$ is not simplified because 12 has a perfect square factor of 4.

Examples

• $\sqrt{20}$ is not simplified because 20 has a perfect square factor of 4. It simplifies to $\sqrt{4 \cdot 5} = 2\sqrt{5}$.

• $\sqrt[3]{54}$ is not simplified because 54 has a perfect cube factor of 27. It simplifies to $\sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2}$.

Property

If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, and $n \geq 2$ is an integer, then

To simplify using this property, find the largest perfect power factor in the radicand, rewrite the radicand as a product, use the rule to separate the radicals, and simplify the root of the perfect power.

Examples

• To simplify $\sqrt{75}$, find the largest perfect square factor, 25. Rewrite as $\sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5\sqrt{3}$.

• To simplify $\sqrt[3]{40x^4}$, find the largest perfect cube factor, $8x^3$. Rewrite as $\sqrt[3]{8x^3 \cdot 5x} = \sqrt[3]{8x^3} \cdot \sqrt[3]{5x} = 2x\sqrt[3]{5x}$.

Property

When simplifying expressions with sums or differences involving radicals, simplify the radical term first. An integer and a radical cannot be combined by addition or subtraction as they are not like terms. For fractions, simplify the radical, then factor the numerator to see if any common factors can be removed from the numerator and denominator.

Examples

• To simplify $5 + \sqrt{32}$, first simplify the radical: $5 + \sqrt{16 \cdot 2} = 5 + 4\sqrt{2}$. The terms cannot be added.

• To simplify $\frac{8 - \sqrt{48}}{4}$, simplify the radical: $\frac{8 - \sqrt{16 \cdot 3}}{4} = \frac{8 - 4\sqrt{3}}{4}$. Then factor the numerator: $\frac{4(2 - \sqrt{3})}{4} = 2 - \sqrt{3}$.

Property

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

When simplifying, always try to simplify the fraction inside the radicand first. If you cannot, use the Quotient Property to split the radical into two, then simplify the numerator and denominator separately.

Examples

• To simplify $\sqrt{\frac{75}{48}}$, first simplify the fraction inside: $\sqrt{\frac{25 \cdot 3}{16 \cdot 3}} = \sqrt{\frac{25}{16}} = \frac{5}{4}$.

• To simplify $\sqrt{\frac{50x^3}{y^6}}$, use the property: $\frac{\sqrt{50x^3}}{\sqrt{y^6}} = \frac{\sqrt{25x^2 \cdot 2x}}{y^3} = \frac{5x\sqrt{2x}}{y^3}$.