Evaluate 1/( square root of 1-(1/2)^2)

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

Step 1

Simplify the denominator.

Apply the product rule to $\frac{1}{2}$ .

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

One to any power is one.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Subtract $1$ from $4$ .

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

Rewrite $\sqrt{\frac{3}{4}}$ as $\frac{\sqrt{3}}{\sqrt{4}}$ .

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

Simplify the denominator.

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

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

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

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{2}{\sqrt{3}}$ by $1$ .

$\frac{2}{\sqrt{3}}$

Step 4

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{2 \sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{3}}{3}$

Decimal Form:

$1.15470053 \dots$

Evaluate 1/( square root of 1-(1/2)^2)

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