Algebra Examples

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}}} - \frac{10 x^{\frac{4}{3}}}{3 {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

To write $\frac{{(5 x^{2} + 1)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{{(5 x^{2} + 1)}^{\frac{2}{3}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$ .

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3}}}{3 x^{\frac{2}{3}}} \cdot \frac{{(5 x^{2} + 1)}^{\frac{2}{3}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}} - \frac{10 x^{\frac{4}{3}}}{3 {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.2

To write $- \frac{10 x^{\frac{4}{3}}}{3 {(5 x^{2} + 1)}^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

Step 1.3

Write each expression with a common denominator of $3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

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

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}} - \frac{10 x^{\frac{4}{3}}}{3 {(5 x^{2} + 1)}^{\frac{2}{3}}} \cdot \frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.3.2

Multiply $\frac{10 x^{\frac{4}{3}}}{3 {(5 x^{2} + 1)}^{\frac{2}{3}}}$ by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}} - \frac{10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 {(5 x^{2} + 1)}^{\frac{2}{3}} x^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.3.3

Reorder the factors of $3 {(5 x^{2} + 1)}^{\frac{2}{3}} x^{\frac{2}{3}}$ .

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}} - \frac{10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}} - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5

Rewrite $\frac{{(5 x^{2} + 1)}^{\frac{1}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}} - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}$ in a factored form.

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Step 1.5.1

Multiply ${(5 x^{2} + 1)}^{\frac{1}{3}}$ by ${(5 x^{2} + 1)}^{\frac{2}{3}}$ by adding the exponents.

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Step 1.5.1.1

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

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1}{3} + \frac{2}{3}} - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.1.2

Combine the numerators over the common denominator.

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{1 + 2}{3}} - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.1.3

Add $1$ and $2$ .

$\frac{\frac{{(5 x^{2} + 1)}^{\frac{3}{3}} - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.1.4

Divide $3$ by $3$ .

$\frac{\frac{{(5 x^{2} + 1)}^{1} - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.2

Simplify ${(5 x^{2} + 1)}^{1}$ .

$\frac{\frac{5 x^{2} + 1 - 10 x^{\frac{4}{3}} x^{\frac{2}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.3

Multiply $x^{\frac{4}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.5.3.1

Move $x^{\frac{2}{3}}$ .

$\frac{\frac{5 x^{2} + 1 - 10 (x^{\frac{2}{3}} x^{\frac{4}{3}})}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.3.2

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

$\frac{\frac{5 x^{2} + 1 - 10 x^{\frac{2}{3} + \frac{4}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.3.3

Combine the numerators over the common denominator.

$\frac{\frac{5 x^{2} + 1 - 10 x^{\frac{2 + 4}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.3.4

Add $2$ and $4$ .

$\frac{\frac{5 x^{2} + 1 - 10 x^{\frac{6}{3}}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.3.5

Divide $6$ by $3$ .

$\frac{\frac{5 x^{2} + 1 - 10 x^{2}}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 1.5.4

Subtract $10 x^{2}$ from $5 x^{2}$ .

$\frac{\frac{- 5 x^{2} + 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}}{{(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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Step 3.1

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

$\frac{- 5 x^{2} + 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}}}$

Step 3.2

Multiply ${(5 x^{2} + 1)}^{\frac{2}{3}}$ by ${(5 x^{2} + 1)}^{\frac{2}{3}}$ by adding the exponents.

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Step 3.2.1

Move ${(5 x^{2} + 1)}^{\frac{2}{3}}$ .

$\frac{- 5 x^{2} + 1}{3 x^{\frac{2}{3}} ({(5 x^{2} + 1)}^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3}})}$

Step 3.2.2

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

$\frac{- 5 x^{2} + 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2}{3} + \frac{2}{3}}}$

Step 3.2.3

Combine the numerators over the common denominator.

$\frac{- 5 x^{2} + 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{2 + 2}{3}}}$

Step 3.2.4

Add $2$ and $2$ .

$\frac{- 5 x^{2} + 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{4}{3}}}$

Step 4

Factor $- 1$ out of $- 5 x^{2}$ .

$\frac{- (5 x^{2}) + 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{4}{3}}}$

Step 5

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (5 x^{2}) - 1 (- 1)}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{4}{3}}}$

Step 6

Factor $- 1$ out of $- (5 x^{2}) - 1 (- 1)$ .

$\frac{- (5 x^{2} - 1)}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{4}{3}}}$

Step 7

Simplify the expression.

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Step 7.1

Rewrite $- (5 x^{2} - 1)$ as $- 1 (5 x^{2} - 1)$ .

$\frac{- 1 (5 x^{2} - 1)}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{4}{3}}}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{5 x^{2} - 1}{3 x^{\frac{2}{3}} {(5 x^{2} + 1)}^{\frac{4}{3}}}$