Trigonometry Examples

$2 {(64)}^{- \frac{1}{2}} + 64^{- \frac{2}{3}}$

Step 1

Simplify each term.

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Step 1.1

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

$2 \frac{1}{64^{\frac{1}{2}}} + 64^{- \frac{2}{3}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$2 \frac{1}{{(8^{2})}^{\frac{1}{2}}} + 64^{- \frac{2}{3}}$

Step 1.2.2

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

$2 \frac{1}{8^{2 (\frac{1}{2})}} + 64^{- \frac{2}{3}}$

Step 1.2.3

Cancel the common factor of $2$ .

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Step 1.2.3.1

Cancel the common factor.

$2 \frac{1}{8^{2 (\frac{1}{2})}} + 64^{- \frac{2}{3}}$

Step 1.2.3.2

Rewrite the expression.

$2 \frac{1}{8^{1}} + 64^{- \frac{2}{3}}$

Step 1.2.4

Evaluate the exponent.

$2 (\frac{1}{8}) + 64^{- \frac{2}{3}}$

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Factor $2$ out of $8$ .

$2 \frac{1}{2 (4)} + 64^{- \frac{2}{3}}$

Step 1.3.2

Cancel the common factor.

$2 \frac{1}{2 \cdot 4} + 64^{- \frac{2}{3}}$

Step 1.3.3

Rewrite the expression.

$\frac{1}{4} + 64^{- \frac{2}{3}}$

Step 1.4

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

$\frac{1}{4} + \frac{1}{64^{\frac{2}{3}}}$

Step 1.5

Simplify the denominator.

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Step 1.5.1

Rewrite $64$ as $4^{3}$ .

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

Step 1.5.2

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

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

Step 1.5.3

Cancel the common factor of $3$ .

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

$\frac{1}{4} + \frac{1}{4^{2}}$

Step 1.5.4

Raise $4$ to the power of $2$ .

$\frac{1}{4} + \frac{1}{16}$

Step 2

To write $\frac{1}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{4} \cdot \frac{4}{4} + \frac{1}{16}$

Step 3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{4}$ by $\frac{4}{4}$ .

$\frac{4}{4 \cdot 4} + \frac{1}{16}$

Step 3.2

Multiply $4$ by $4$ .

$\frac{4}{16} + \frac{1}{16}$

Step 4

Combine the numerators over the common denominator.

$\frac{4 + 1}{16}$

Step 5

Add $4$ and $1$ .

$\frac{5}{16}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{16}$

Decimal Form:

$0.3125$