Algebra Examples

$\frac{1}{\sqrt{1 - {(\frac{\sqrt{3}}{2})}^{2}}}$

Step 1

Simplify the denominator.

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Step 1.1

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\sqrt{1 - \frac{{\sqrt{3}}^{2}}{2^{2}}}}$

Step 1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{1}{\sqrt{1 - \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}}$

Step 1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\sqrt{1 - \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}}$

Step 1.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{1}{\sqrt{1 - \frac{3^{\frac{2}{2}}}{2^{2}}}}$

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

$\frac{1}{\sqrt{1 - \frac{3^{\frac{2}{2}}}{2^{2}}}}$

Step 1.2.4.2

Rewrite the expression.

$\frac{1}{\sqrt{1 - \frac{3^{1}}{2^{2}}}}$

Step 1.2.5

Evaluate the exponent.

$\frac{1}{\sqrt{1 - \frac{3}{2^{2}}}}$

Step 1.3

Raise $2$ to the power of $2$ .

$\frac{1}{\sqrt{1 - \frac{3}{4}}}$

Step 1.4

Write $1$ as a fraction with a common denominator.

$\frac{1}{\sqrt{\frac{4}{4} - \frac{3}{4}}}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{4 - 3}{4}}}$

Step 1.6

Subtract $3$ from $4$ .

$\frac{1}{\sqrt{\frac{1}{4}}}$

Step 1.7

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$\frac{1}{\frac{\sqrt{1}}{\sqrt{4}}}$

Step 1.8

Any root of $1$ is $1$ .

$\frac{1}{\frac{1}{\sqrt{4}}}$

Step 1.9

Simplify the denominator.

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Step 1.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.9.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{1}{\frac{1}{2}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 3

Multiply $2$ by $1$ .

$2$