Evaluate (4/3)/( square root of 9-(4/3)^2)

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{4}{3} \cdot \frac{1}{\sqrt{9 - {(\frac{4}{3})}^{2}}}$

Step 2

Combine fractions.

Combine.

$\frac{4 \cdot 1}{3 \sqrt{9 - {(\frac{4}{3})}^{2}}}$

Multiply $4$ by $1$ .

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

Step 3

Simplify the denominator.

Apply the product rule to $\frac{4}{3}$ .

$\frac{4}{3 \sqrt{9 - \frac{4^{2}}{3^{2}}}}$

Raise $4$ to the power of $2$ .

$\frac{4}{3 \sqrt{9 - \frac{16}{3^{2}}}}$

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

$\frac{4}{3 \sqrt{9 - \frac{16}{9}}}$

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

$\frac{4}{3 \sqrt{9 \cdot \frac{9}{9} - \frac{16}{9}}}$

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

$\frac{4}{3 \sqrt{\frac{9 \cdot 9}{9} - \frac{16}{9}}}$

Combine the numerators over the common denominator.

$\frac{4}{3 \sqrt{\frac{9 \cdot 9 - 16}{9}}}$

Simplify the numerator.

Multiply $9$ by $9$ .

$\frac{4}{3 \sqrt{\frac{81 - 16}{9}}}$

Subtract $16$ from $81$ .

$\frac{4}{3 \sqrt{\frac{65}{9}}}$

Rewrite $\sqrt{\frac{65}{9}}$ as $\frac{\sqrt{65}}{\sqrt{9}}$ .

$\frac{4}{3 \frac{\sqrt{65}}{\sqrt{9}}}$

Simplify the denominator.

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

$\frac{4}{3 \frac{\sqrt{65}}{\sqrt{3^{2}}}}$

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

$\frac{4}{3 \frac{\sqrt{65}}{3}}$

Step 4

Combine $3$ and $\frac{\sqrt{65}}{3}$ .

$\frac{4}{\frac{3 \sqrt{65}}{3}}$

Step 5

Reduce the expression by cancelling the common factors.

Reduce the expression $\frac{3 \sqrt{65}}{3}$ by cancelling the common factors.

Cancel the common factor.

$\frac{4}{\frac{3 \sqrt{65}}{3}}$

Rewrite the expression.

$\frac{4}{\frac{\sqrt{65}}{1}}$

Divide $\sqrt{65}$ by $1$ .

$\frac{4}{\sqrt{65}}$

Step 6

Multiply $\frac{4}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\frac{4}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}}$

Step 7

Combine and simplify the denominator.

Multiply $\frac{4}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\frac{4 \sqrt{65}}{\sqrt{65} \sqrt{65}}$

Raise $\sqrt{65}$ to the power of $1$ .

$\frac{4 \sqrt{65}}{{\sqrt{65}}^{1} \sqrt{65}}$

Raise $\sqrt{65}$ to the power of $1$ .

$\frac{4 \sqrt{65}}{{\sqrt{65}}^{1} {\sqrt{65}}^{1}}$

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

$\frac{4 \sqrt{65}}{{\sqrt{65}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{4 \sqrt{65}}{{\sqrt{65}}^{2}}$

Rewrite ${\sqrt{65}}^{2}$ as $65$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{65}$ as $65^{\frac{1}{2}}$ .

$\frac{4 \sqrt{65}}{{(65^{\frac{1}{2}})}^{2}}$

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

$\frac{4 \sqrt{65}}{65^{\frac{1}{2} \cdot 2}}$

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

$\frac{4 \sqrt{65}}{65^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{4 \sqrt{65}}{65^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{4 \sqrt{65}}{65^{1}}$

Evaluate the exponent.

$\frac{4 \sqrt{65}}{65}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{4 \sqrt{65}}{65}$

Decimal Form:

$0.49613893 \dots$

Evaluate (4/3)/( square root of 9-(4/3)^2)

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