Precalculus Examples

$\ln ({(\frac{x + 7}{{(x^{2} - 49)}^{5}})}^{\frac{3}{5}})$

Step 1

Simplify the denominator.

Tap for more steps...

Step 1.1

Rewrite $49$ as $7^{2}$ .

$\ln ({(\frac{x + 7}{{(x^{2} - 7^{2})}^{5}})}^{\frac{3}{5}})$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 7$ .

$\ln ({(\frac{x + 7}{{((x + 7) (x - 7))}^{5}})}^{\frac{3}{5}})$

Step 1.3

Apply the product rule to $(x + 7) (x - 7)$ .

$\ln ({(\frac{x + 7}{{(x + 7)}^{5} {(x - 7)}^{5}})}^{\frac{3}{5}})$

Step 2

Cancel the common factor of $x + 7$ and ${(x + 7)}^{5}$ .

Tap for more steps...

Step 2.1

Multiply by $1$ .

$\ln ({(\frac{(x + 7) \cdot 1}{{(x + 7)}^{5} {(x - 7)}^{5}})}^{\frac{3}{5}})$

Step 2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1

Factor $x + 7$ out of ${(x + 7)}^{5} {(x - 7)}^{5}$ .

$\ln ({(\frac{(x + 7) \cdot 1}{(x + 7) ({(x + 7)}^{4} {(x - 7)}^{5})})}^{\frac{3}{5}})$

Step 2.2.2

Cancel the common factor.

$\ln ({(\frac{(x + 7) \cdot 1}{(x + 7) ({(x + 7)}^{4} {(x - 7)}^{5})})}^{\frac{3}{5}})$

Step 2.2.3

Rewrite the expression.

$\ln ({(\frac{1}{{(x + 7)}^{4} {(x - 7)}^{5}})}^{\frac{3}{5}})$

Step 3

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

Tap for more steps...

Step 3.1

Apply the product rule to $\frac{1}{{(x + 7)}^{4} {(x - 7)}^{5}}$ .

$\ln (\frac{1^{\frac{3}{5}}}{{({(x + 7)}^{4} {(x - 7)}^{5})}^{\frac{3}{5}}})$

Step 3.2

Apply the product rule to ${(x + 7)}^{4} {(x - 7)}^{5}$ .

$\ln (\frac{1^{\frac{3}{5}}}{{({(x + 7)}^{4})}^{\frac{3}{5}} {({(x - 7)}^{5})}^{\frac{3}{5}}})$

Step 4

One to any power is one.

$\ln (\frac{1}{{({(x + 7)}^{4})}^{\frac{3}{5}} {({(x - 7)}^{5})}^{\frac{3}{5}}})$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Multiply the exponents in ${({(x + 7)}^{4})}^{\frac{3}{5}}$ .

Tap for more steps...

Step 5.1.1

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

$\ln (\frac{1}{{(x + 7)}^{4 (\frac{3}{5})} {({(x - 7)}^{5})}^{\frac{3}{5}}})$

Step 5.1.2

Multiply $4 (\frac{3}{5})$ .

Tap for more steps...

Step 5.1.2.1

Combine $4$ and $\frac{3}{5}$ .

$\ln (\frac{1}{{(x + 7)}^{\frac{4 \cdot 3}{5}} {({(x - 7)}^{5})}^{\frac{3}{5}}})$

Step 5.1.2.2

Multiply $4$ by $3$ .

$\ln (\frac{1}{{(x + 7)}^{\frac{12}{5}} {({(x - 7)}^{5})}^{\frac{3}{5}}})$

Step 5.2

Multiply the exponents in ${({(x - 7)}^{5})}^{\frac{3}{5}}$ .

Tap for more steps...

Step 5.2.1

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

$\ln (\frac{1}{{(x + 7)}^{\frac{12}{5}} {(x - 7)}^{5 (\frac{3}{5})}})$

Step 5.2.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 5.2.2.1

Cancel the common factor.

$\ln (\frac{1}{{(x + 7)}^{\frac{12}{5}} {(x - 7)}^{5 (\frac{3}{5})}})$

Step 5.2.2.2

Rewrite the expression.

$\ln (\frac{1}{{(x + 7)}^{\frac{12}{5}} {(x - 7)}^{3}})$