Basic Math Examples

${(\frac{4 \cdot 32^{\frac{3}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1

Simplify the numerator.

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Step 1.1

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

${(\frac{2^{2} \cdot 32^{\frac{3}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.2

Rewrite $32$ as $2^{5}$ .

${(\frac{2^{2} \cdot {(2^{5})}^{\frac{3}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.3

Multiply the exponents in ${(2^{5})}^{\frac{3}{2}}$ .

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Step 1.3.1

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

${(\frac{2^{2} \cdot 2^{5 (\frac{3}{2})}}{128^{\frac{1}{2}}})}^{12}$

Step 1.3.2

Multiply $5 (\frac{3}{2})$ .

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Step 1.3.2.1

Combine $5$ and $\frac{3}{2}$ .

${(\frac{2^{2} \cdot 2^{\frac{5 \cdot 3}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.3.2.2

Multiply $5$ by $3$ .

${(\frac{2^{2} \cdot 2^{\frac{15}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.4

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

${(\frac{2^{2 + \frac{15}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.5

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

${(\frac{2^{2 \cdot \frac{2}{2} + \frac{15}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.6

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

${(\frac{2^{\frac{2 \cdot 2}{2} + \frac{15}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.7

Combine the numerators over the common denominator.

${(\frac{2^{\frac{2 \cdot 2 + 15}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.8

Simplify the numerator.

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Step 1.8.1

Multiply $2$ by $2$ .

${(\frac{2^{\frac{4 + 15}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 1.8.2

Add $4$ and $15$ .

${(\frac{2^{\frac{19}{2}}}{128^{\frac{1}{2}}})}^{12}$

Step 2

Apply basic rules of exponents.

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Step 2.1

Apply the product rule to $\frac{2^{\frac{19}{2}}}{128^{\frac{1}{2}}}$ .

$\frac{{(2^{\frac{19}{2}})}^{12}}{{(128^{\frac{1}{2}})}^{12}}$

Step 2.2

Multiply the exponents in ${(2^{\frac{19}{2}})}^{12}$ .

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Step 2.2.1

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

$\frac{2^{\frac{19}{2} \cdot 12}}{{(128^{\frac{1}{2}})}^{12}}$

Step 2.2.2

Cancel the common factor of $2$ .

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Step 2.2.2.1

Factor $2$ out of $12$ .

$\frac{2^{\frac{19}{2} \cdot (2 (6))}}{{(128^{\frac{1}{2}})}^{12}}$

Step 2.2.2.2

Cancel the common factor.

$\frac{2^{\frac{19}{2} \cdot (2 \cdot 6)}}{{(128^{\frac{1}{2}})}^{12}}$

Step 2.2.2.3

Rewrite the expression.

$\frac{2^{19 \cdot 6}}{{(128^{\frac{1}{2}})}^{12}}$

Step 2.2.3

Multiply $19$ by $6$ .

$\frac{2^{114}}{{(128^{\frac{1}{2}})}^{12}}$

Step 3

Simplify the denominator.

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Step 3.1

Multiply the exponents in ${(128^{\frac{1}{2}})}^{12}$ .

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Step 3.1.1

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

$\frac{2^{114}}{128^{\frac{1}{2} \cdot 12}}$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Factor $2$ out of $12$ .

$\frac{2^{114}}{128^{\frac{1}{2} \cdot (2 (6))}}$

Step 3.1.2.2

Cancel the common factor.

$\frac{2^{114}}{128^{\frac{1}{2} \cdot (2 \cdot 6)}}$

Step 3.1.2.3

Rewrite the expression.

$\frac{2^{114}}{128^{6}}$

Step 3.2

Raise $128$ to the power of $6$ .

$\frac{2^{114}}{4398046511104}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{2^{114}}{4398046511104}$

Decimal Form:

$4.72236648 \cdot 10^{21}$