Basic Math Examples

${(\frac{2}{{(\frac{16}{81})}^{- \frac{3}{4}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1

Simplify the denominator.

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Step 1.1

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

${(\frac{2}{{(\frac{81}{16})}^{\frac{3}{4}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.2

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

${(\frac{2}{\frac{81^{\frac{3}{4}}}{16^{\frac{3}{4}}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

Rewrite $81$ as $3^{4}$ .

${(\frac{2}{\frac{{(3^{4})}^{\frac{3}{4}}}{16^{\frac{3}{4}}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.3.2

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

${(\frac{2}{\frac{3^{4 (\frac{3}{4})}}{16^{\frac{3}{4}}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.3.3

Cancel the common factor of $4$ .

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Step 1.3.3.1

Cancel the common factor.

${(\frac{2}{\frac{3^{4 (\frac{3}{4})}}{16^{\frac{3}{4}}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.3.3.2

Rewrite the expression.

${(\frac{2}{\frac{3^{3}}{16^{\frac{3}{4}}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.3.4

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

${(\frac{2}{\frac{27}{16^{\frac{3}{4}}} + {(\frac{9}{4})}^{- \frac{1}{2}}})}^{- 2}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

Cancel the common factor of $4$ .

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Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

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

Step 1.4.4

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

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

Step 1.5

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

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

Step 1.6

Apply the product rule to $\frac{4}{9}$ .

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

Step 1.7

Simplify the numerator.

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Step 1.7.1

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

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

Step 1.7.2

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

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

Step 1.7.3

Cancel the common factor of $2$ .

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Step 1.7.3.1

Cancel the common factor.

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

Step 1.7.3.2

Rewrite the expression.

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

Step 1.7.4

Evaluate the exponent.

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

Step 1.8

Simplify the denominator.

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Step 1.8.1

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

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

Step 1.8.2

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

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

Step 1.8.3

Cancel the common factor of $2$ .

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Step 1.8.3.1

Cancel the common factor.

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

Step 1.8.3.2

Rewrite the expression.

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

Step 1.8.4

Evaluate the exponent.

${(\frac{2}{\frac{27}{8} + \frac{2}{3}})}^{- 2}$

Step 1.9

To write $\frac{27}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

${(\frac{2}{\frac{27}{8} \cdot \frac{3}{3} + \frac{2}{3}})}^{- 2}$

Step 1.10

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

${(\frac{2}{\frac{27}{8} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{8}{8}})}^{- 2}$

Step 1.11

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.11.1

Multiply $\frac{27}{8}$ by $\frac{3}{3}$ .

${(\frac{2}{\frac{27 \cdot 3}{8 \cdot 3} + \frac{2}{3} \cdot \frac{8}{8}})}^{- 2}$

Step 1.11.2

Multiply $8$ by $3$ .

${(\frac{2}{\frac{27 \cdot 3}{24} + \frac{2}{3} \cdot \frac{8}{8}})}^{- 2}$

Step 1.11.3

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

${(\frac{2}{\frac{27 \cdot 3}{24} + \frac{2 \cdot 8}{3 \cdot 8}})}^{- 2}$

Step 1.11.4

Multiply $3$ by $8$ .

${(\frac{2}{\frac{27 \cdot 3}{24} + \frac{2 \cdot 8}{24}})}^{- 2}$

Step 1.12

Combine the numerators over the common denominator.

${(\frac{2}{\frac{27 \cdot 3 + 2 \cdot 8}{24}})}^{- 2}$

Step 1.13

Simplify the numerator.

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Step 1.13.1

Multiply $27$ by $3$ .

${(\frac{2}{\frac{81 + 2 \cdot 8}{24}})}^{- 2}$

Step 1.13.2

Multiply $2$ by $8$ .

${(\frac{2}{\frac{81 + 16}{24}})}^{- 2}$

Step 1.13.3

Add $81$ and $16$ .

${(\frac{2}{\frac{97}{24}})}^{- 2}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

${(2 (\frac{24}{97}))}^{- 2}$

Step 3

Multiply $2 (\frac{24}{97})$ .

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Step 3.1

Combine $2$ and $\frac{24}{97}$ .

${(\frac{2 \cdot 24}{97})}^{- 2}$

Step 3.2

Multiply $2$ by $24$ .

${(\frac{48}{97})}^{- 2}$

Step 4

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

${(\frac{97}{48})}^{2}$

Step 5

Apply the product rule to $\frac{97}{48}$ .

$\frac{97^{2}}{48^{2}}$

Step 6

Raise $97$ to the power of $2$ .

$\frac{9409}{48^{2}}$

Step 7

Raise $48$ to the power of $2$ .

$\frac{9409}{2304}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{9409}{2304}$

Decimal Form:

$4.08376736 \dots$

Mixed Number Form:

$4 \frac{193}{2304}$