Simplify (6(7+x)^(1/3)-x(7+x)^(-2/3))/((7+x)^(2/3))

$\frac{6 {(7 + x)}^{\frac{1}{3}} - x {(7 + x)}^{- \frac{2}{3}}}{{(7 + x)}^{\frac{2}{3}}}$

Simplify the numerator.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{6 {(7 + x)}^{\frac{1}{3}} - x \frac{1}{{(7 + x)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Combine $\frac{1}{{(7 + x)}^{\frac{2}{3}}}$ and $x$ .

$\frac{6 {(7 + x)}^{\frac{1}{3}} - \frac{x}{{(7 + x)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

To write $6 {(7 + x)}^{\frac{1}{3}}$ as a fraction with a common denominator, multiply by $\frac{{(7 + x)}^{\frac{2}{3}}}{{(7 + x)}^{\frac{2}{3}}}$ .

$\frac{\frac{6 {(7 + x)}^{\frac{1}{3}} {(7 + x)}^{\frac{2}{3}}}{{(7 + x)}^{\frac{2}{3}}} - \frac{x}{{(7 + x)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Combine the numerators over the common denominator.

$\frac{\frac{6 {(7 + x)}^{\frac{1}{3}} {(7 + x)}^{\frac{2}{3}} - x}{{(7 + x)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Reorder terms.

$\frac{\frac{6 {(7 + x)}^{\frac{2}{3}} {(7 + x)}^{\frac{1}{3}} - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Rewrite $\frac{6 {(7 + x)}^{\frac{2}{3}} {(7 + x)}^{\frac{1}{3}} - x}{{(x + 7)}^{\frac{2}{3}}}$ in a factored form.

Multiply ${(7 + x)}^{\frac{2}{3}}$ by ${(7 + x)}^{\frac{1}{3}}$ by adding the exponents.

Move ${(7 + x)}^{\frac{1}{3}}$ .

$\frac{\frac{6 ({(7 + x)}^{\frac{1}{3}} {(7 + x)}^{\frac{2}{3}}) - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

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

$\frac{\frac{6 {(7 + x)}^{\frac{1}{3} + \frac{2}{3}} - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Combine the numerators over the common denominator.

$\frac{\frac{6 {(7 + x)}^{\frac{1 + 2}{3}} - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Add $1$ and $2$ .

$\frac{\frac{6 {(7 + x)}^{\frac{3}{3}} - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Divide $3$ by $3$ .

$\frac{\frac{6 {(7 + x)}^{1} - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Simplify $6 {(7 + x)}^{1}$ .

$\frac{\frac{6 (7 + x) - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Apply the distributive property.

$\frac{\frac{6 \cdot 7 + 6 x - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Multiply $6$ by $7$ .

$\frac{\frac{42 + 6 x - x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Subtract $x$ from $6 x$ .

$\frac{\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3}}}}{{(7 + x)}^{\frac{2}{3}}}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3}}} \cdot \frac{1}{{(7 + x)}^{\frac{2}{3}}}$

Multiply $\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3}}} \cdot \frac{1}{{(7 + x)}^{\frac{2}{3}}}$ .

Multiply $\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3}}}$ by $\frac{1}{{(7 + x)}^{\frac{2}{3}}}$ .

$\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3}} {(7 + x)}^{\frac{2}{3}}}$

Reorder terms.

$\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3}} {(x + 7)}^{\frac{2}{3}}}$

Multiply ${(x + 7)}^{\frac{2}{3}}$ by ${(x + 7)}^{\frac{2}{3}}$ by adding the exponents.

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

$\frac{42 + 5 x}{{(x + 7)}^{\frac{2}{3} + \frac{2}{3}}}$

Combine the numerators over the common denominator.

$\frac{42 + 5 x}{{(x + 7)}^{\frac{2 + 2}{3}}}$

Add $2$ and $2$ .

$\frac{42 + 5 x}{{(x + 7)}^{\frac{4}{3}}}$

Simplify (6(7+x)^(1/3)-x(7+x)^(-2/3))/((7+x)^(2/3))

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