Basic Math Examples

${(\frac{125}{8})}^{- \frac{2}{3}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1

Simplify each term.

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Step 1.1

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{8}{125})}^{\frac{2}{3}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.2

Apply the product rule to $\frac{8}{125}$ .

$\frac{8^{\frac{2}{3}}}{125^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

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

$\frac{{(2^{3})}^{\frac{2}{3}}}{125^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.3.2

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

$\frac{2^{3 (\frac{2}{3})}}{125^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.3.3

Cancel the common factor of $3$ .

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Step 1.3.3.1

Cancel the common factor.

$\frac{2^{3 (\frac{2}{3})}}{125^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.3.3.2

Rewrite the expression.

$\frac{2^{2}}{125^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.3.4

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

$\frac{4}{125^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

Rewrite $125$ as $5^{3}$ .

$\frac{4}{{(5^{3})}^{\frac{2}{3}}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.4.2

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

$\frac{4}{5^{3 (\frac{2}{3})}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.4.3

Cancel the common factor of $3$ .

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Step 1.4.3.1

Cancel the common factor.

$\frac{4}{5^{3 (\frac{2}{3})}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.4.3.2

Rewrite the expression.

$\frac{4}{5^{2}} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.4.4

Raise $5$ to the power of $2$ .

$\frac{4}{25} + {(\frac{81}{64})}^{- \frac{3}{2}}$

Step 1.5

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{4}{25} + {(\frac{64}{81})}^{\frac{3}{2}}$

Step 1.6

Apply the product rule to $\frac{64}{81}$ .

$\frac{4}{25} + \frac{64^{\frac{3}{2}}}{81^{\frac{3}{2}}}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Rewrite $64$ as $8^{2}$ .

$\frac{4}{25} + \frac{{(8^{2})}^{\frac{3}{2}}}{81^{\frac{3}{2}}}$

Step 1.7.2

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

$\frac{4}{25} + \frac{8^{2 (\frac{3}{2})}}{81^{\frac{3}{2}}}$

Step 1.7.3

Cancel the common factor of $2$ .

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Step 1.7.3.1

Cancel the common factor.

$\frac{4}{25} + \frac{8^{2 (\frac{3}{2})}}{81^{\frac{3}{2}}}$

Step 1.7.3.2

Rewrite the expression.

$\frac{4}{25} + \frac{8^{3}}{81^{\frac{3}{2}}}$

Step 1.7.4

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

$\frac{4}{25} + \frac{512}{81^{\frac{3}{2}}}$

Step 1.8

Simplify the denominator.

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Step 1.8.1

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

$\frac{4}{25} + \frac{512}{{(9^{2})}^{\frac{3}{2}}}$

Step 1.8.2

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

$\frac{4}{25} + \frac{512}{9^{2 (\frac{3}{2})}}$

Step 1.8.3

Cancel the common factor of $2$ .

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Step 1.8.3.1

Cancel the common factor.

$\frac{4}{25} + \frac{512}{9^{2 (\frac{3}{2})}}$

Step 1.8.3.2

Rewrite the expression.

$\frac{4}{25} + \frac{512}{9^{3}}$

Step 1.8.4

Raise $9$ to the power of $3$ .

$\frac{4}{25} + \frac{512}{729}$

Step 2

To write $\frac{4}{25}$ as a fraction with a common denominator, multiply by $\frac{729}{729}$ .

$\frac{4}{25} \cdot \frac{729}{729} + \frac{512}{729}$

Step 3

To write $\frac{512}{729}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$\frac{4}{25} \cdot \frac{729}{729} + \frac{512}{729} \cdot \frac{25}{25}$

Step 4

Write each expression with a common denominator of $18225$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4}{25}$ by $\frac{729}{729}$ .

$\frac{4 \cdot 729}{25 \cdot 729} + \frac{512}{729} \cdot \frac{25}{25}$

Step 4.2

Multiply $25$ by $729$ .

$\frac{4 \cdot 729}{18225} + \frac{512}{729} \cdot \frac{25}{25}$

Step 4.3

Multiply $\frac{512}{729}$ by $\frac{25}{25}$ .

$\frac{4 \cdot 729}{18225} + \frac{512 \cdot 25}{729 \cdot 25}$

Step 4.4

Multiply $729$ by $25$ .

$\frac{4 \cdot 729}{18225} + \frac{512 \cdot 25}{18225}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 \cdot 729 + 512 \cdot 25}{18225}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $4$ by $729$ .

$\frac{2916 + 512 \cdot 25}{18225}$

Step 6.2

Multiply $512$ by $25$ .

$\frac{2916 + 12800}{18225}$

Step 6.3

Add $2916$ and $12800$ .

$\frac{15716}{18225}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{15716}{18225}$

Decimal Form:

$0.86233196 \dots$