Precalculus Examples

$\frac{{(4 s^{- 6} t^{4})}^{\frac{3}{2}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(4 \frac{1}{s^{6}} t^{4})}^{\frac{3}{2}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.2

Combine $4$ and $\frac{1}{s^{6}}$ .

$\frac{{(\frac{4}{s^{6}} t^{4})}^{\frac{3}{2}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.3

Combine $\frac{4}{s^{6}}$ and $t^{4}$ .

$\frac{{(\frac{4 t^{4}}{s^{6}})}^{\frac{3}{2}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.4

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

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Step 1.4.1

Apply the product rule to $\frac{4 t^{4}}{s^{6}}$ .

$\frac{\frac{{(4 t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.4.2

Apply the product rule to $4 t^{4}$ .

$\frac{\frac{4^{\frac{3}{2}} {(t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{{(2^{2})}^{\frac{3}{2}} {(t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.2

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

$\frac{\frac{2^{2 (\frac{3}{2})} {(t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.3

Cancel the common factor of $2$ .

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Step 1.5.3.1

Cancel the common factor.

$\frac{\frac{2^{2 (\frac{3}{2})} {(t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.3.2

Rewrite the expression.

$\frac{\frac{2^{3} {(t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.4

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

$\frac{\frac{8 {(t^{4})}^{\frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.5

Multiply the exponents in ${(t^{4})}^{\frac{3}{2}}$ .

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Step 1.5.5.1

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

$\frac{\frac{8 t^{4 (\frac{3}{2})}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.5.2

Cancel the common factor of $2$ .

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Step 1.5.5.2.1

Factor $2$ out of $4$ .

$\frac{\frac{8 t^{2 (2) \frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.5.2.2

Cancel the common factor.

$\frac{\frac{8 t^{2 \cdot 2 \frac{3}{2}}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.5.2.3

Rewrite the expression.

$\frac{\frac{8 t^{2 \cdot 3}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5.5.3

Multiply $2$ by $3$ .

$\frac{\frac{8 t^{6}}{{(s^{6})}^{\frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.6

Multiply the exponents in ${(s^{6})}^{\frac{3}{2}}$ .

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Step 1.6.1

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

$\frac{\frac{8 t^{6}}{s^{6 (\frac{3}{2})}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.6.2

Cancel the common factor of $2$ .

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Step 1.6.2.1

Factor $2$ out of $6$ .

$\frac{\frac{8 t^{6}}{s^{2 (3) \frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.6.2.2

Cancel the common factor.

$\frac{\frac{8 t^{6}}{s^{2 \cdot 3 \frac{3}{2}}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.6.2.3

Rewrite the expression.

$\frac{\frac{8 t^{6}}{s^{3 \cdot 3}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 1.6.3

Multiply $3$ by $3$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{{(8 s^{4} t^{- 4})}^{\frac{2}{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{8 t^{6}}{s^{9}}}{{(8 s^{4} \frac{1}{t^{4}})}^{\frac{2}{3}}}$

Step 2.2

Multiply $8 s^{4} \frac{1}{t^{4}}$ .

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Step 2.2.1

Combine $8$ and $\frac{1}{t^{4}}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{{(s^{4} \frac{8}{t^{4}})}^{\frac{2}{3}}}$

Step 2.2.2

Combine $s^{4}$ and $\frac{8}{t^{4}}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{{(\frac{s^{4} \cdot 8}{t^{4}})}^{\frac{2}{3}}}$

Step 2.3

Move $8$ to the left of $s^{4}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{{(\frac{8 \cdot s^{4}}{t^{4}})}^{\frac{2}{3}}}$

Step 2.4

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

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Step 2.4.1

Apply the product rule to $\frac{8 s^{4}}{t^{4}}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{{(8 s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.4.2

Apply the product rule to $8 s^{4}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{8^{\frac{2}{3}} {(s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Rewrite $8$ as $2^{3}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{{(2^{3})}^{\frac{2}{3}} {(s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.2

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

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{2^{3 (\frac{2}{3})} {(s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.3

Cancel the common factor of $3$ .

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Step 2.5.3.1

Cancel the common factor.

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{2^{3 (\frac{2}{3})} {(s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.3.2

Rewrite the expression.

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{2^{2} {(s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.4

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

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 {(s^{4})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.5

Multiply the exponents in ${(s^{4})}^{\frac{2}{3}}$ .

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Step 2.5.5.1

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

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 s^{4 (\frac{2}{3})}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.5.2

Multiply $4 (\frac{2}{3})$ .

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Step 2.5.5.2.1

Combine $4$ and $\frac{2}{3}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 s^{\frac{4 \cdot 2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.5.2.2

Multiply $4$ by $2$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 s^{\frac{8}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.6

Multiply the exponents in ${(t^{4})}^{\frac{2}{3}}$ .

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Step 2.6.1

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

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 s^{\frac{8}{3}}}{t^{4 (\frac{2}{3})}}}$

Step 2.6.2

Multiply $4 (\frac{2}{3})$ .

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Step 2.6.2.1

Combine $4$ and $\frac{2}{3}$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 s^{\frac{8}{3}}}{t^{\frac{4 \cdot 2}{3}}}}$

Step 2.6.2.2

Multiply $4$ by $2$ .

$\frac{\frac{8 t^{6}}{s^{9}}}{\frac{4 s^{\frac{8}{3}}}{t^{\frac{8}{3}}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 t^{6}}{s^{9}} \cdot \frac{t^{\frac{8}{3}}}{4 s^{\frac{8}{3}}}$

Step 4

Combine.

$\frac{8 t^{6} t^{\frac{8}{3}}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5

Multiply $t^{6}$ by $t^{\frac{8}{3}}$ by adding the exponents.

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Step 5.1

Move $t^{\frac{8}{3}}$ .

$\frac{8 (t^{\frac{8}{3}} t^{6})}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5.2

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

$\frac{8 t^{\frac{8}{3} + 6}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5.3

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8 t^{\frac{8}{3} + 6 \cdot \frac{3}{3}}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5.4

Combine $6$ and $\frac{3}{3}$ .

$\frac{8 t^{\frac{8}{3} + \frac{6 \cdot 3}{3}}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5.5

Combine the numerators over the common denominator.

$\frac{8 t^{\frac{8 + 6 \cdot 3}{3}}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply $6$ by $3$ .

$\frac{8 t^{\frac{8 + 18}{3}}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 5.6.2

Add $8$ and $18$ .

$\frac{8 t^{\frac{26}{3}}}{s^{9} (4 s^{\frac{8}{3}})}$

Step 6

Multiply $s^{9}$ by $s^{\frac{8}{3}}$ by adding the exponents.

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Step 6.1

Move $s^{\frac{8}{3}}$ .

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{8}{3}} s^{9} \cdot 4}$

Step 6.2

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

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{8}{3} + 9} \cdot 4}$

Step 6.3

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{8}{3} + 9 \cdot \frac{3}{3}} \cdot 4}$

Step 6.4

Combine $9$ and $\frac{3}{3}$ .

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{8}{3} + \frac{9 \cdot 3}{3}} \cdot 4}$

Step 6.5

Combine the numerators over the common denominator.

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{8 + 9 \cdot 3}{3}} \cdot 4}$

Step 6.6

Simplify the numerator.

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Step 6.6.1

Multiply $9$ by $3$ .

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{8 + 27}{3}} \cdot 4}$

Step 6.6.2

Add $8$ and $27$ .

$\frac{8 t^{\frac{26}{3}}}{s^{\frac{35}{3}} \cdot 4}$

Step 7

Factor $4$ out of $8 t^{\frac{26}{3}}$ .

$\frac{4 (2 t^{\frac{26}{3}})}{s^{\frac{35}{3}} \cdot 4}$

Step 8

Cancel the common factors.

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Step 8.1

Factor $4$ out of $s^{\frac{35}{3}} \cdot 4$ .

$\frac{4 (2 t^{\frac{26}{3}})}{4 \cdot s^{\frac{35}{3}}}$

Step 8.2

Cancel the common factor.

$\frac{4 (2 t^{\frac{26}{3}})}{4 \cdot s^{\frac{35}{3}}}$

Step 8.3

Rewrite the expression.

$\frac{2 t^{\frac{26}{3}}}{s^{\frac{35}{3}}}$