Algebra Examples

$\frac{4^{- 5}}{4^{7}} \cdot 4^{- 7}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{\frac{1}{4^{5}}}{4^{7}} \cdot 4^{- 7}$

Step 1.2

Raise $4$ to the power of $5$ .

$\frac{\frac{1}{1024}}{4^{7}} \cdot 4^{- 7}$

Step 2

Raise $4$ to the power of $7$ .

$\frac{\frac{1}{1024}}{16384} \cdot 4^{- 7}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{1024} \cdot \frac{1}{16384} \cdot 4^{- 7}$

Step 4

Multiply $\frac{1}{1024} \cdot \frac{1}{16384}$ .

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Step 4.1

Multiply $\frac{1}{1024}$ by $\frac{1}{16384}$ .

$\frac{1}{1024 \cdot 16384} \cdot 4^{- 7}$

Step 4.2

Multiply $1024$ by $16384$ .

$\frac{1}{16777216} \cdot 4^{- 7}$

Step 5

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

$\frac{1}{16777216} \cdot \frac{1}{4^{7}}$

Step 6

Combine fractions.

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Step 6.1

Combine.

$\frac{1 \cdot 1}{16777216 \cdot 4^{7}}$

Step 6.2

Multiply $1$ by $1$ .

$\frac{1}{16777216 \cdot 4^{7}}$

Step 7

Simplify the denominator.

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Step 7.1

Rewrite $16777216$ as $2^{24}$ .

$\frac{1}{2^{24} \cdot 4^{7}}$

Step 7.2

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

$\frac{1}{2^{24} \cdot {(2^{2})}^{7}}$

Step 7.3

Multiply the exponents in ${(2^{2})}^{7}$ .

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Step 7.3.1

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

$\frac{1}{2^{24} \cdot 2^{2 \cdot 7}}$

Step 7.3.2

Multiply $2$ by $7$ .

$\frac{1}{2^{24} \cdot 2^{14}}$

Step 7.4

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

$\frac{1}{2^{24 + 14}}$

Step 7.5

Add $24$ and $14$ .

$\frac{1}{2^{38}}$

Step 8

Raise $2$ to the power of $38$ .

$\frac{1}{274877906944}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{274877906944}$

Decimal Form:

$3.6379788 \cdot 10^{- 12}$