Reducing Radicals with Variables

​Reducing Radicals with Variables

- When reducing radicals, we look for perfect squares.

- With variables, a perfect square is when there is an even exponent.

For Example:

These are all perfect squares!

​Reducing Radicals with Variables

- When reducing radicals that are perfect squares, we can pull the variable out by dividing the exponent by 2.

For Example:

Notice how there is not a "√" sign anymore.

Another Example:

Reduce each radical:

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​Let's Practice That

Multiple Choice

Reduce the expression x⁶.

• $$x^2$$

• $$x^3$$

• $$x^6$$

• $$\sqrt[]{x^3}$$

Multiple Choice

Reduce the expression x¹².

• $$x^6$$

• $$x^8$$

• $$x^{12}$$

• $$\sqrt[]{x^6}$$

Multiple Choice

Reduce the expression x⁸.

• $$\sqrt[]{x^2}$$

• $$x^2$$

• $$\sqrt[]{x^4}$$

• $$x^4$$

Math Response

Reduce: $$\sqrt[]{x^{18}}$$

Type answer here

Correct Answer: x^9

Deg^°

Rad

​Reducing Radicals with Variables

- If a varialbe is not a perfect squares, you can always pull one out to make a perfect square!

For Example:

This is not a perfect square.

But, I can pull an "x" out to create a perfect square!

"x⁶" is a perfect square!

Rewrite each expression so that it becomes a perfect square:

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​Let's Practice That

Multiple Choice

Rewrite x⁹ so that it is a perfect square:

• $$x^8$$

• $$x^8\cdot x$$

• $$x^7\cdot x^2$$

• It is already a perfect square.

Multiple Choice

Rewrite x⁵ so that it is a perfect square:

• $$x^4$$

• $$x^5\cdot x$$

• $$x^4\cdot x$$

• It is already a perfect square.

Multiple Choice

Rewrite x¹¹ so that it is a perfect square:

• $$x^{10}\cdot x$$

• $$x^{11}\cdot x$$

• $$x^6\cdot x^5$$

• It is already a perfect square.

Multiple Choice

Rewrite x¹⁴ so that it is a perfect square:

• $$x^{13}\cdot x$$

• $$x^{14}\cdot x$$

• $$x^7\cdot x^7$$

• It is already a perfect square.

​Reducing Radicals with Variables

- Once your variable is split up to create a perfect square, you can reduce!

For Example:

- "x⁶" is a perfect square so it can be reduced to x³.

Rewrite each expression so that it becomes a perfect square and then reduce:

​

​Let's Practice That

Multiple Choice

Reduce: $$\sqrt[]{x^5}$$

• $$x^4\sqrt[]{x}$$

• $$x\sqrt[]{x^2}$$

• $$x\sqrt[]{x^4}$$

• $$x^2\sqrt[]{x}$$

Multiple Choice

Reduce: $$\sqrt[]{x^9}$$

• $$x^4\sqrt[]{x}$$

• $$x\sqrt[]{x^8}$$

• $$x\sqrt[]{x^4}$$

• $$x^8\sqrt[]{x}$$

Multiple Choice

Reduce: $$\sqrt[]{x^{21}}$$

• $$x^{20}\sqrt[]{x}$$

• $$x\sqrt[]{x^{10}}$$

• $$x^{10}\sqrt[]{x}$$

• $$x^{20}\sqrt[]{x}$$

Multiple Choice

Reduce: $$\sqrt[]{x^{17}}$$

• $$x^4\sqrt[]{x}$$

• $$x\sqrt[]{x^{16}}$$

• $$x^8\sqrt[]{x}$$

• $$x\sqrt[]{x^8}$$

Multiple Choice

Reduce: $$\sqrt[]{x^3}$$

• $$x\sqrt[]{x}$$

• $$x^2\sqrt[]{x}$$

• $$x^3\sqrt[]{x}$$

• $$x\sqrt[]{x^3}$$