Basic Math Examples

${(- \frac{27}{8})}^{- 3} \div {(\frac{81}{16})}^{- \frac{1}{2}}$

Step 1

Rewrite the division as a fraction.

$\frac{{(- \frac{27}{8})}^{- 3}}{{(\frac{81}{16})}^{- \frac{1}{2}}}$

Step 2

Simplify the expression.

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Step 2.1

Move ${(- \frac{27}{8})}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(\frac{81}{16})}^{- \frac{1}{2}} {(- \frac{27}{8})}^{3}}$

Step 2.2

Move ${(\frac{81}{16})}^{- \frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(\frac{81}{16})}^{\frac{1}{2}}}{{(- \frac{27}{8})}^{3}}$

Step 3

Simplify the numerator.

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Step 3.1

Apply the product rule to $\frac{81}{16}$ .

$\frac{\frac{81^{\frac{1}{2}}}{16^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $81$ as $9^{2}$ .

$\frac{\frac{{(9^{2})}^{\frac{1}{2}}}{16^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.2.2

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

$\frac{\frac{9^{2 (\frac{1}{2})}}{16^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.2.3

Cancel the common factor of $2$ .

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Step 3.2.3.1

Cancel the common factor.

$\frac{\frac{9^{2 (\frac{1}{2})}}{16^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.2.3.2

Rewrite the expression.

$\frac{\frac{9^{1}}{16^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.2.4

Evaluate the exponent.

$\frac{\frac{9}{16^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $16$ as $4^{2}$ .

$\frac{\frac{9}{{(4^{2})}^{\frac{1}{2}}}}{{(- \frac{27}{8})}^{3}}$

Step 3.3.2

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

$\frac{\frac{9}{4^{2 (\frac{1}{2})}}}{{(- \frac{27}{8})}^{3}}$

Step 3.3.3

Cancel the common factor of $2$ .

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Step 3.3.3.1

Cancel the common factor.

$\frac{\frac{9}{4^{2 (\frac{1}{2})}}}{{(- \frac{27}{8})}^{3}}$

Step 3.3.3.2

Rewrite the expression.

$\frac{\frac{9}{4^{1}}}{{(- \frac{27}{8})}^{3}}$

Step 3.3.4

Evaluate the exponent.

$\frac{\frac{9}{4}}{{(- \frac{27}{8})}^{3}}$

Step 4

Simplify the denominator.

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Step 4.1

Apply the product rule to $- \frac{27}{8}$ .

$\frac{\frac{9}{4}}{{(- 1)}^{3} {(\frac{27}{8})}^{3}}$

Step 4.2

Raise $- 1$ to the power of $3$ .

$\frac{\frac{9}{4}}{- {(\frac{27}{8})}^{3}}$

Step 4.3

Apply the product rule to $\frac{27}{8}$ .

$\frac{\frac{9}{4}}{- \frac{27^{3}}{8^{3}}}$

Step 4.4

Raise $27$ to the power of $3$ .

$\frac{\frac{9}{4}}{- \frac{19683}{8^{3}}}$

Step 4.5

Raise $8$ to the power of $3$ .

$\frac{\frac{9}{4}}{- \frac{19683}{512}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{9}{4} (- \frac{512}{19683})$

Step 6

Cancel the common factor of $9$ .

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Step 6.1

Move the leading negative in $- \frac{512}{19683}$ into the numerator.

$\frac{9}{4} \cdot \frac{- 512}{19683}$

Step 6.2

Factor $9$ out of $19683$ .

$\frac{9}{4} \cdot \frac{- 512}{9 (2187)}$

Step 6.3

Cancel the common factor.

$\frac{9}{4} \cdot \frac{- 512}{9 \cdot 2187}$

Step 6.4

Rewrite the expression.

$\frac{1}{4} \cdot \frac{- 512}{2187}$

Step 7

Cancel the common factor of $4$ .

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Step 7.1

Factor $4$ out of $- 512$ .

$\frac{1}{4} \cdot \frac{4 (- 128)}{2187}$

Step 7.2

Cancel the common factor.

$\frac{1}{4} \cdot \frac{4 \cdot - 128}{2187}$

Step 7.3

Rewrite the expression.

$\frac{- 128}{2187}$

Step 8

Move the negative in front of the fraction.

$- \frac{128}{2187}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{128}{2187}$

Decimal Form:

$- 0.05852766 \dots$