Algebra Examples

\frac{\sqrt{a}}{\sqrt{b}} = x

$\frac{\sqrt{a}}{\sqrt{b}} = x$

Step 1

Cross multiply.

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Step 1.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

x \cdot (\sqrt{b}) = \sqrt{a}

$x \cdot (\sqrt{b}) = \sqrt{a}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Multiply

x

$x$ by

\sqrt{b}

$\sqrt{b}$ .

x \sqrt{b} = \sqrt{a}

$x \sqrt{b} = \sqrt{a}$

x \sqrt{b} = \sqrt{a}

$x \sqrt{b} = \sqrt{a}$

x \sqrt{b} = \sqrt{a}

$x \sqrt{b} = \sqrt{a}$

Step 2

Rewrite the equation as

\sqrt{a} = x \sqrt{b}

$\sqrt{a} = x \sqrt{b}$ .

\sqrt{a} = x \sqrt{b}

$\sqrt{a} = x \sqrt{b}$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

{\sqrt{a}}^{2} = {(x \sqrt{b})}^{2}

${\sqrt{a}}^{2} = {(x \sqrt{b})}^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{a}

$\sqrt{a}$ as

a^{\frac{1}{2}}

$a^{\frac{1}{2}}$ .

{(a^{\frac{1}{2}})}^{2} = {(x \sqrt{b})}^{2}

${(a^{\frac{1}{2}})}^{2} = {(x \sqrt{b})}^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify

{(a^{\frac{1}{2}})}^{2}

${(a^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply the exponents in

{(a^{\frac{1}{2}})}^{2}

${(a^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

a^{\frac{1}{2} \cdot 2} = {(x \sqrt{b})}^{2}

$a^{\frac{1}{2} \cdot 2} = {(x \sqrt{b})}^{2}$

Step 4.2.1.1.2

Cancel the common factor of

2

$2$ .

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Step 4.2.1.1.2.1

Cancel the common factor.

$a^{\frac{1}{2} \cdot 2} = {(x \sqrt{b})}^{2}$

Step 4.2.1.1.2.2

Rewrite the expression.

$a^{1} = {(x \sqrt{b})}^{2}$

Step 4.2.1.2

Simplify.

$a = {(x \sqrt{b})}^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify

${(x \sqrt{b})}^{2}$ .

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Step 4.3.1.1

Apply the product rule to

$x \sqrt{b}$ .

$a = x^{2} {\sqrt{b}}^{2}$

Step 4.3.1.2

Rewrite

${\sqrt{b}}^{2}$ as

$b$ .

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Step 4.3.1.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{b}$ as

$b^{\frac{1}{2}}$ .

$a = x^{2} {(b^{\frac{1}{2}})}^{2}$

Step 4.3.1.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$a = x^{2} b^{\frac{1}{2} \cdot 2}$

Step 4.3.1.2.3

Combine

$\frac{1}{2}$ and

$2$ .

$a = x^{2} b^{\frac{2}{2}}$

Step 4.3.1.2.4

Cancel the common factor of

$2$ .

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Step 4.3.1.2.4.1

Cancel the common factor.

$a = x^{2} b^{\frac{2}{2}}$

Step 4.3.1.2.4.2

Rewrite the expression.

$a = x^{2} b^{1}$

Step 4.3.1.2.5

Simplify.

$a = x^{2} b$