Precalculus Examples

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

Step 1

Combine $2$ and $\frac{4}{\sqrt{65}}$ .

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

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Combine and simplify the denominator.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.2.6

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

Tap for more steps...

Step 2.2.6.1

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

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

Step 2.2.6.2

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

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.6.4.1

Cancel the common factor.

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

Step 2.2.6.4.2

Rewrite the expression.

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

Step 2.2.6.5

Evaluate the exponent.

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

Step 2.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 2.3.1

Apply the product rule to $\frac{4 \sqrt{65}}{65}$ .

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

Step 2.3.2

Apply the product rule to $4 \sqrt{65}$ .

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

Step 2.4

Simplify the numerator.

Tap for more steps...

Step 2.4.1

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

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 {\sqrt{65}}^{2}}{65^{2}}}$

Step 2.4.2

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

Tap for more steps...

Step 2.4.2.1

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

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 {(65^{\frac{1}{2}})}^{2}}{65^{2}}}$

Step 2.4.2.2

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

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 \cdot 65^{\frac{1}{2} \cdot 2}}{65^{2}}}$

Step 2.4.2.3

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

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 \cdot 65^{\frac{2}{2}}}{65^{2}}}$

Step 2.4.2.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.4.2.4.1

Cancel the common factor.

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 \cdot 65^{\frac{2}{2}}}{65^{2}}}$

Step 2.4.2.4.2

Rewrite the expression.

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 \cdot 65^{1}}{65^{2}}}$

Step 2.4.2.5

Evaluate the exponent.

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 \cdot 65}{65^{2}}}$

Step 2.5

Raise $65$ to the power of $2$ .

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16 \cdot 65}{4225}}$

Step 2.6

Multiply $16$ by $65$ .

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{1040}{4225}}$

Step 2.7

Cancel the common factor of $1040$ and $4225$ .

Tap for more steps...

Step 2.7.1

Factor $65$ out of $1040$ .

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{65 (16)}{4225}}$

Step 2.7.2

Cancel the common factors.

Tap for more steps...

Step 2.7.2.1

Factor $65$ out of $4225$ .

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{65 \cdot 16}{65 \cdot 65}}$

Step 2.7.2.2

Cancel the common factor.

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{65 \cdot 16}{65 \cdot 65}}$

Step 2.7.2.3

Rewrite the expression.

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{1 - \frac{16}{65}}$

Step 2.8

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

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{\frac{65}{65} - \frac{16}{65}}$

Step 2.9

Combine the numerators over the common denominator.

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{\frac{65 - 16}{65}}$

Step 2.10

Subtract $16$ from $65$ .

$\frac{\frac{2 \cdot 4}{\sqrt{65}}}{\frac{49}{65}}$

Step 3

Multiply $2$ by $4$ .

$\frac{\frac{8}{\sqrt{65}}}{\frac{49}{65}}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Multiply $\frac{8}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\frac{\frac{8}{\sqrt{65}} \cdot \frac{\sqrt{65}}{\sqrt{65}}}{\frac{49}{65}}$

Step 4.2

Combine and simplify the denominator.

Tap for more steps...

Step 4.2.1

Multiply $\frac{8}{\sqrt{65}}$ by $\frac{\sqrt{65}}{\sqrt{65}}$ .

$\frac{\frac{8 \sqrt{65}}{\sqrt{65} \sqrt{65}}}{\frac{49}{65}}$

Step 4.2.2

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

$\frac{\frac{8 \sqrt{65}}{{\sqrt{65}}^{1} \sqrt{65}}}{\frac{49}{65}}$

Step 4.2.3

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

$\frac{\frac{8 \sqrt{65}}{{\sqrt{65}}^{1} {\sqrt{65}}^{1}}}{\frac{49}{65}}$

Step 4.2.4

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

$\frac{\frac{8 \sqrt{65}}{{\sqrt{65}}^{1 + 1}}}{\frac{49}{65}}$

Step 4.2.5

Add $1$ and $1$ .

$\frac{\frac{8 \sqrt{65}}{{\sqrt{65}}^{2}}}{\frac{49}{65}}$

Step 4.2.6

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

Tap for more steps...

Step 4.2.6.1

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

$\frac{\frac{8 \sqrt{65}}{{(65^{\frac{1}{2}})}^{2}}}{\frac{49}{65}}$

Step 4.2.6.2

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

$\frac{\frac{8 \sqrt{65}}{65^{\frac{1}{2} \cdot 2}}}{\frac{49}{65}}$

Step 4.2.6.3

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

$\frac{\frac{8 \sqrt{65}}{65^{\frac{2}{2}}}}{\frac{49}{65}}$

Step 4.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.2.6.4.1

Cancel the common factor.

$\frac{\frac{8 \sqrt{65}}{65^{\frac{2}{2}}}}{\frac{49}{65}}$

Step 4.2.6.4.2

Rewrite the expression.

$\frac{\frac{8 \sqrt{65}}{65^{1}}}{\frac{49}{65}}$

Step 4.2.6.5

Evaluate the exponent.

$\frac{\frac{8 \sqrt{65}}{65}}{\frac{49}{65}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 \sqrt{65}}{65} \cdot \frac{65}{49}$

Step 6

Cancel the common factor of $65$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

$\frac{8 \sqrt{65}}{65} \cdot \frac{65}{49}$

Step 6.2

Rewrite the expression.

$8 \sqrt{65} \frac{1}{49}$

Step 7

Combine $\frac{1}{49}$ and $8$ .

$\frac{8}{49} \sqrt{65}$

Step 8

Combine $\frac{8}{49}$ and $\sqrt{65}$ .

$\frac{8 \sqrt{65}}{49}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{8 \sqrt{65}}{49}$

Decimal Form:

$1.31628697 \dots$