Algebra Examples

$g (\frac{2}{3}) = \frac{x}{\sqrt{9 - x^{2}}}$

Step 1

Replace the variable $x$ with $\frac{2}{3}$ in the expression.

$g (\frac{2}{3}) = \frac{\frac{2}{3}}{\sqrt{9 - {(\frac{2}{3})}^{2}}}$

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$g (\frac{2}{3}) = \frac{2}{3} \cdot \frac{1}{\sqrt{9 - {(\frac{2}{3})}^{2}}}$

Step 2.2

Combine fractions.

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Step 2.2.1

Combine.

$g (\frac{2}{3}) = \frac{2 \cdot 1}{3 \sqrt{9 - {(\frac{2}{3})}^{2}}}$

Step 2.2.2

Multiply $2$ by $1$ .

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{9 - {(\frac{2}{3})}^{2}}}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

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

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{9 - \frac{2^{2}}{3^{2}}}}$

Step 2.3.2

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

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{9 - \frac{4}{3^{2}}}}$

Step 2.3.3

Raise $3$ to the power of $2$ .

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{9 - \frac{4}{9}}}$

Step 2.3.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{9 \cdot \frac{9}{9} - \frac{4}{9}}}$

Step 2.3.5

Combine $9$ and $\frac{9}{9}$ .

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{\frac{9 \cdot 9}{9} - \frac{4}{9}}}$

Step 2.3.6

Combine the numerators over the common denominator.

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{\frac{9 \cdot 9 - 4}{9}}}$

Step 2.3.7

Simplify the numerator.

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Step 2.3.7.1

Multiply $9$ by $9$ .

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{\frac{81 - 4}{9}}}$

Step 2.3.7.2

Subtract $4$ from $81$ .

$g (\frac{2}{3}) = \frac{2}{3 \sqrt{\frac{77}{9}}}$

Step 2.3.8

Rewrite $\sqrt{\frac{77}{9}}$ as $\frac{\sqrt{77}}{\sqrt{9}}$ .

$g (\frac{2}{3}) = \frac{2}{3 (\frac{\sqrt{77}}{\sqrt{9}})}$

Step 2.3.9

Simplify the denominator.

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Step 2.3.9.1

Rewrite $9$ as $3^{2}$ .

$g (\frac{2}{3}) = \frac{2}{3 (\frac{\sqrt{77}}{\sqrt{3^{2}}})}$

Step 2.3.9.2

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

$g (\frac{2}{3}) = \frac{2}{3 (\frac{\sqrt{77}}{3})}$

Step 2.4

Combine $3$ and $\frac{\sqrt{77}}{3}$ .

$g (\frac{2}{3}) = \frac{2}{\frac{3 \sqrt{77}}{3}}$

Step 2.5

Reduce the expression by cancelling the common factors.

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Step 2.5.1

Reduce the expression $\frac{3 \sqrt{77}}{3}$ by cancelling the common factors.

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Step 2.5.1.1

Cancel the common factor.

$g (\frac{2}{3}) = \frac{2}{\frac{3 \sqrt{77}}{3}}$

Step 2.5.1.2

Rewrite the expression.

$g (\frac{2}{3}) = \frac{2}{\frac{\sqrt{77}}{1}}$

Step 2.5.2

Divide $\sqrt{77}$ by $1$ .

$g (\frac{2}{3}) = \frac{2}{\sqrt{77}}$

Step 2.6

Multiply $\frac{2}{\sqrt{77}}$ by $\frac{\sqrt{77}}{\sqrt{77}}$ .

$g (\frac{2}{3}) = \frac{2}{\sqrt{77}} \cdot \frac{\sqrt{77}}{\sqrt{77}}$

Step 2.7

Combine and simplify the denominator.

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Step 2.7.1

Multiply $\frac{2}{\sqrt{77}}$ by $\frac{\sqrt{77}}{\sqrt{77}}$ .

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{\sqrt{77} \sqrt{77}}$

Step 2.7.2

Raise $\sqrt{77}$ to the power of $1$ .

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{\sqrt{77} \sqrt{77}}$

Step 2.7.3

Raise $\sqrt{77}$ to the power of $1$ .

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{\sqrt{77} \sqrt{77}}$

Step 2.7.4

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

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{{\sqrt{77}}^{1 + 1}}$

Step 2.7.5

Add $1$ and $1$ .

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{{\sqrt{77}}^{2}}$

Step 2.7.6

Rewrite ${\sqrt{77}}^{2}$ as $77$ .

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Step 2.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{77}$ as $77^{\frac{1}{2}}$ .

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{{(77^{\frac{1}{2}})}^{2}}$

Step 2.7.6.2

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

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{77^{\frac{1}{2} \cdot 2}}$

Step 2.7.6.3

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

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{77^{\frac{2}{2}}}$

Step 2.7.6.4

Cancel the common factor of $2$ .

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Step 2.7.6.4.1

Cancel the common factor.

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{77^{\frac{2}{2}}}$

Step 2.7.6.4.2

Rewrite the expression.

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{77}$

Step 2.7.6.5

Evaluate the exponent.

$g (\frac{2}{3}) = \frac{2 \sqrt{77}}{77}$

Step 2.8

The final answer is $\frac{2 \sqrt{77}}{77}$ .

$\frac{2 \sqrt{77}}{77}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{77}}{77}$

Decimal Form:

$0.22792115 \dots$

Step 4