Precalculus Examples

$\frac{{(9 s t)}^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

Apply the product rule to $9 s t$ .

$\frac{{(9 s)}^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1.1.2

Apply the product rule to $9 s$ .

$\frac{9^{\frac{3}{2}} s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$\frac{{(3^{2})}^{\frac{3}{2}} s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1.3

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

$\frac{3^{2 (\frac{3}{2})} s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\frac{3^{2 (\frac{3}{2})} s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1.4.2

Rewrite the expression.

$\frac{3^{3} s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 1.5

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

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} t^{- 4})}^{\frac{2}{3}}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(27 s^{3} \frac{1}{t^{4}})}^{\frac{2}{3}}}$

Step 2.2

Multiply $27 s^{3} \frac{1}{t^{4}}$ .

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Step 2.2.1

Combine $27$ and $\frac{1}{t^{4}}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(s^{3} \frac{27}{t^{4}})}^{\frac{2}{3}}}$

Step 2.2.2

Combine $s^{3}$ and $\frac{27}{t^{4}}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(\frac{s^{3} \cdot 27}{t^{4}})}^{\frac{2}{3}}}$

Step 2.3

Move $27$ to the left of $s^{3}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{{(\frac{27 \cdot s^{3}}{t^{4}})}^{\frac{2}{3}}}$

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 $\frac{27 s^{3}}{t^{4}}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{{(27 s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.4.2

Apply the product rule to $27 s^{3}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{27^{\frac{2}{3}} {(s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Rewrite $27$ as $3^{3}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{{(3^{3})}^{\frac{2}{3}} {(s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.2

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

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{3^{3 (\frac{2}{3})} {(s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.3

Cancel the common factor of $3$ .

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Step 2.5.3.1

Cancel the common factor.

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{3^{3 (\frac{2}{3})} {(s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.3.2

Rewrite the expression.

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{3^{2} {(s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.4

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

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 {(s^{3})}^{\frac{2}{3}}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.5

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

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Step 2.5.5.1

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

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 s^{3 (\frac{2}{3})}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.5.2

Cancel the common factor of $3$ .

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Step 2.5.5.2.1

Cancel the common factor.

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 s^{3 (\frac{2}{3})}}{{(t^{4})}^{\frac{2}{3}}}}$

Step 2.5.5.2.2

Rewrite the expression.

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 s^{2}}{{(t^{4})}^{\frac{2}{3}}}}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 s^{2}}{t^{4 (\frac{2}{3})}}}$

Step 2.6.2

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

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Step 2.6.2.1

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

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 s^{2}}{t^{\frac{4 \cdot 2}{3}}}}$

Step 2.6.2.2

Multiply $4$ by $2$ .

$\frac{27 s^{\frac{3}{2}} t^{\frac{3}{2}}}{\frac{9 s^{2}}{t^{\frac{8}{3}}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$27 s^{\frac{3}{2}} t^{\frac{3}{2}} \frac{t^{\frac{8}{3}}}{9 s^{2}}$

Step 4

Cancel the common factor of $s^{\frac{3}{2}} \cdot 9$ .

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Step 4.1

Factor $s^{\frac{3}{2}} \cdot 9$ out of $27 s^{\frac{3}{2}} t^{\frac{3}{2}}$ .

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

Step 4.2

Factor $s^{\frac{3}{2}} \cdot 9$ out of $9 s^{2}$ .

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

Step 4.3

Cancel the common factor.

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

Step 4.4

Rewrite the expression.

$3 t^{\frac{3}{2}} \frac{t^{\frac{8}{3}}}{s^{\frac{1}{2}}}$

Step 5

Combine $3$ and $\frac{t^{\frac{8}{3}}}{s^{\frac{1}{2}}}$ .

$t^{\frac{3}{2}} \frac{3 t^{\frac{8}{3}}}{s^{\frac{1}{2}}}$

Step 6

Combine $t^{\frac{3}{2}}$ and $\frac{3 t^{\frac{8}{3}}}{s^{\frac{1}{2}}}$ .

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

Step 7

Multiply $t^{\frac{3}{2}}$ by $t^{\frac{8}{3}}$ by adding the exponents.

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Step 7.1

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

$\frac{t^{\frac{8}{3}} t^{\frac{3}{2}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.2

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

$\frac{t^{\frac{8}{3} + \frac{3}{2}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.3

To write $\frac{8}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{t^{\frac{8}{3} \cdot \frac{2}{2} + \frac{3}{2}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.4

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{t^{\frac{8}{3} \cdot \frac{2}{2} + \frac{3}{2} \cdot \frac{3}{3}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.5.1

Multiply $\frac{8}{3}$ by $\frac{2}{2}$ .

$\frac{t^{\frac{8 \cdot 2}{3 \cdot 2} + \frac{3}{2} \cdot \frac{3}{3}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.5.2

Multiply $3$ by $2$ .

$\frac{t^{\frac{8 \cdot 2}{6} + \frac{3}{2} \cdot \frac{3}{3}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.5.3

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$\frac{t^{\frac{8 \cdot 2}{6} + \frac{3 \cdot 3}{2 \cdot 3}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.5.4

Multiply $2$ by $3$ .

$\frac{t^{\frac{8 \cdot 2}{6} + \frac{3 \cdot 3}{6}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.6

Combine the numerators over the common denominator.

$\frac{t^{\frac{8 \cdot 2 + 3 \cdot 3}{6}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.7

Simplify the numerator.

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Step 7.7.1

Multiply $8$ by $2$ .

$\frac{t^{\frac{16 + 3 \cdot 3}{6}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.7.2

Multiply $3$ by $3$ .

$\frac{t^{\frac{16 + 9}{6}} \cdot 3}{s^{\frac{1}{2}}}$

Step 7.7.3

Add $16$ and $9$ .

$\frac{t^{\frac{25}{6}} \cdot 3}{s^{\frac{1}{2}}}$

Step 8

Move $3$ to the left of $t^{\frac{25}{6}}$ .

$\frac{3 t^{\frac{25}{6}}}{s^{\frac{1}{2}}}$