Simplifying Radicals With Variables, Exponents, Fractions, Cube ... - Free Printable

To simplify the given radicals, we will break down each expression step by step. Let's go through each one:

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1. Simplify \( \sqrt{32x^2} \)

#### Step 1: Factorize the radicand
The radicand is \( 32x^2 \). We can factorize it as follows:
- \( 32 = 2^5 \)
- \( x^2 \) is already in its simplest form.

So, \( 32x^2 = 2^5 \cdot x^2 \).

#### Step 2: Apply the square root
The square root of a product is the product of the square roots:
\[
\sqrt{32x^2} = \sqrt{2^5 \cdot x^2} = \sqrt{2^5} \cdot \sqrt{x^2}
\]

- For \( \sqrt{2^5} \):
\[
\sqrt{2^5} = \sqrt{2^4 \cdot 2} = \sqrt{2^4} \cdot \sqrt{2} = 2^2 \cdot \sqrt{2} = 4\sqrt{2}
\]

- For \( \sqrt{x^2} \):
\[
\sqrt{x^2} = |x|
\]

#### Step 3: Combine the results
\[
\sqrt{32x^2} = 4\sqrt{2} \cdot |x| = 4|x|\sqrt{2}
\]

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2. Simplify \( \sqrt{18x^5y^3} \)

#### Step 1: Factorize the radicand
The radicand is \( 18x^5y^3 \). We can factorize it as follows:
- \( 18 = 2 \cdot 3^2 \)
- \( x^5 = x^4 \cdot x \)
- \( y^3 = y^2 \cdot y \)

So, \( 18x^5y^3 = 2 \cdot 3^2 \cdot x^4 \cdot x \cdot y^2 \cdot y \).

#### Step 2: Apply the square root
The square root of a product is the product of the square roots:
\[
\sqrt{18x^5y^3} = \sqrt{2 \cdot 3^2 \cdot x^4 \cdot x \cdot y^2 \cdot y}
\]

- For \( \sqrt{3^2} \):
\[
\sqrt{3^2} = 3
\]

- For \( \sqrt{x^4} \):
\[
\sqrt{x^4} = x^2
\]

- For \( \sqrt{y^2} \):
\[
\sqrt{y^2} = |y|
\]

- For \( \sqrt{2 \cdot x \cdot y} \):
\[
\sqrt{2 \cdot x \cdot y} = \sqrt{2xy}
\]

#### Step 3: Combine the results
\[
\sqrt{18x^5y^3} = 3 \cdot x^2 \cdot |y| \cdot \sqrt{2xy} = 3x^2|y|\sqrt{2xy}
\]

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3. Simplify \( \sqrt[3]{x^5y^9z^{14}} \)

#### Step 1: Factorize the radicand
The radicand is \( x^5y^9z^{14} \). We can factorize it as follows:
- \( x^5 = x^3 \cdot x^2 \)
- \( y^9 = (y^3)^3 \)
- \( z^{14} = z^{12} \cdot z^2 = (z^4)^3 \cdot z^2 \)

So, \( x^5y^9z^{14} = x^3 \cdot x^2 \cdot (y^3)^3 \cdot (z^4)^3 \cdot z^2 \).

#### Step 2: Apply the cube root
The cube root of a product is the product of the cube roots:
\[
\sqrt[3]{x^5y^9z^{14}} = \sqrt[3]{x^3 \cdot x^2 \cdot (y^3)^3 \cdot (z^4)^3 \cdot z^2}
\]

- For \( \sqrt[3]{x^3} \):
\[
\sqrt[3]{x^3} = x
\]

- For \( \sqrt[3]{(y^3)^3} \):
\[
\sqrt[3]{(y^3)^3} = y^3
\]

- For \( \sqrt[3]{(z^4)^3} \):
\[
\sqrt[3]{(z^4)^3} = z^4
\]

- For \( \sqrt[3]{x^2z^2} \):
\[
\sqrt[3]{x^2z^2} = \sqrt[3]{x^2z^2}
\]

#### Step 3: Combine the results
\[
\sqrt[3]{x^5y^9z^{14}} = x \cdot y^3 \cdot z^4 \cdot \sqrt[3]{x^2z^2} = xy^3z^4\sqrt[3]{x^2z^2}
\]

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4. Simplify \( \sqrt[3]{8a^3b^2} \)

#### Step 1: Factorize the radicand
The radicand is \( 8a^3b^2 \). We can factorize it as follows:
- \( 8 = 2^3 \)
- \( a^3 \) is already a perfect cube.
- \( b^2 \) remains as it is.

So, \( 8a^3b^2 = 2^3 \cdot a^3 \cdot b^2 \).

#### Step 2: Apply the cube root
The cube root of a product is the product of the cube roots:
\[
\sqrt[3]{8a^3b^2} = \sqrt[3]{2^3 \cdot a^3 \cdot b^2}
\]

- For \( \sqrt[3]{2^3} \):
\[
\sqrt[3]{2^3} = 2
\]

- For \( \sqrt[3]{a^3} \):
\[
\sqrt[3]{a^3} = a
\]

- For \( \sqrt[3]{b^2} \):
\[
\sqrt[3]{b^2} = \sqrt[3]{b^2}
\]

#### Step 3: Combine the results
\[
\sqrt[3]{8a^3b^2} = 2 \cdot a \cdot \sqrt[3]{b^2} = 2a\sqrt[3]{b^2}
\]

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Final Answers

\[
\boxed{
4|x|\sqrt{2}, \quad 3x^2|y|\sqrt{2xy}, \quad xy^3z^4\sqrt[3]{x^2z^2}, \quad 2a\sqrt[3]{b^2}
}
\]