Algebra 2 Assignment Simplify Radicals

The properties of exponents, which we've talked about earlier, tell us among other things that

$$\begin{pmatrix} xy \end{pmatrix}^{a}=x^{a}y^{a}$$

$$\begin{pmatrix} \frac{x}{y} \end{pmatrix}^{a}=\frac{x^{a}}{y^{a}} $$

We also know that

$$\sqrt[a]{x}=x^{\frac{1}{a}}$$

$$or$$

$$\sqrt{x}=x^{\frac{1}{2}}$$

If we combine these two things then we get the product property of radicals and the quotient property of radicals. These two properties tell us that the square root of a product equals the product of the square roots of the factors.

$$\sqrt{xy}=\sqrt{x}\cdot \sqrt{y}$$

$$\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$$

$$where\:\: x\geq 0,y\geq 0 $$

The answer can't be negative and x and y can't be negative since we then wouldn't get a real answer. In the same way we know that

$$\sqrt{x^{2}}=x\: \: where\: \: x\geq 0$$

These properties can be used to simplify radical expressions. A radical expression is said to be in its simplest form if there are

no perfect square factors other than 1 in the radicand

$$\sqrt{16x}=\sqrt{16}\cdot \sqrt{x}=\sqrt{4^{2}}\cdot \sqrt{x}=4\sqrt{x}$$

no fractions in the radicand and

$$\sqrt{\frac{25}{16}x^{2}}=\frac{\sqrt{25}}{\sqrt{16}}\cdot \sqrt{x^{2}}=\frac{5}{4}x$$

no radicals appear in the denominator of a fraction.

$$\sqrt{\frac{15}{16}}=\frac{\sqrt{15}}{\sqrt{16}}=\frac{\sqrt{15}}{4}$$

If the denominator is not a perfect square you can rationalize the denominator by multiplying the expression by an appropriate form of 1 e.g.

$$\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}\cdot {\color{green} {\frac{\sqrt{y}}{\sqrt{y}}}}=\frac{\sqrt{xy}}{\sqrt{y^{2}}}=\frac{\sqrt{xy}}{y}$$

Binomials like

$$x\sqrt{y}+z\sqrt{w}\: \: and\: \: x\sqrt{y}-z\sqrt{w}$$

are called conjugates to each other. The product of two conjugates is always a rational number which means that you can use conjugates to rationalize the denominator e.g.

$$\frac{x}{4+\sqrt{x}}=\frac{x\left ( {\color{green} {4-\sqrt{x}}} \right )}{\left ( 4+\sqrt{x} \right )\left ( {\color{green}{ 4-\sqrt{x}}} \right )}= $$

$$=\frac{x\left ( 4-\sqrt{x} \right )}{16-\left ( \sqrt{x} \right )^{2}}=\frac{4x-x\sqrt{x}}{16-x}$$


Video lesson

Simplify the radical expression

$$\frac{x}{5-\sqrt{x}}$$

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Algebra 2 Worksheets

Radical Functions Worksheets

Here is a graphic preview for all of the Radical Functions Worksheets. You can select different variables to customize these Radical Functions Worksheets for your needs. The Radical Functions Worksheets are randomly created and will never repeat so you have an endless supply of quality Radical Functions Worksheets to use in the classroom or at home. Our Radical Functions Worksheets are free to download, easy to use, and very flexible.

These Radical Functions Worksheets are a good resource for students in the 8th Grade through the 12th Grade.

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Simplifying Radicals Worksheets
This radical functions worksheet will produce problems for simplifying radical functions. You may select the degree of the root function and whether to include variables or not. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Operations with Radical Expressions Worksheets
This radical functions worksheet will produce problems for Operations with Radical Functions. You may select the difficulty of the problems. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Dividing Radical Expressions Worksheets
This radical functions worksheet will produce problems for dividing radical expressions. You may select the difficulty of the problems. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Radical and Rational Exponents Worksheets
This radical functions worksheet will produce problems for radical and rational exponents. You may select whether to include variables or not. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Simplifying Rational Exponents Worksheets
This radical functions worksheet will produce problems for simplifying rational exponents. You may select the difficulty of the problems. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Square Root Equations Worksheets
This radical functions worksheet will produce problems for square root equations. You may select the difficulty of the problems. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Rational Exponent Equations Worksheets
This radical functions worksheet will produce problems for rational exponent equations. You may select the difficulty of the problems. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.

Graphing Radicals Worksheets
This radical functions worksheet will produce problems for graphing radicals. You may select the degree of the root. This radical functions worksheet is a good resource for students in the 9th Grade, 10th Grade, 11th Grade, and 12th Grade.


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