Basic Math Examples

${(\frac{4 m^{5}}{3})}^{3} \cdot \frac{{(2 n)}^{3}}{64 m^{8}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $2 n$ .

${(\frac{4 m^{5}}{3})}^{3} \cdot \frac{2^{3} n^{3}}{64 m^{8}}$

Step 1.2

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

${(\frac{4 m^{5}}{3})}^{3} \cdot \frac{8 n^{3}}{64 m^{8}}$

Step 2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1

Apply the product rule to $\frac{4 m^{5}}{3}$ .

$\frac{{(4 m^{5})}^{3}}{3^{3}} \cdot \frac{8 n^{3}}{64 m^{8}}$

Step 2.2

Apply the product rule to $4 m^{5}$ .

$\frac{4^{3} {(m^{5})}^{3}}{3^{3}} \cdot \frac{8 n^{3}}{64 m^{8}}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{4^{3} {(m^{5})}^{3} (8 n^{3})}{3^{3} (64 m^{8})}$

Step 3.2

Cancel the common factor of $8$ and $64$ .

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Step 3.2.1

Factor $8$ out of $4^{3} {(m^{5})}^{3} (8 n^{3})$ .

$\frac{8 (4^{3} {(m^{5})}^{3} (n^{3}))}{3^{3} (64 m^{8})}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $8$ out of $3^{3} (64 m^{8})$ .

$\frac{8 (4^{3} {(m^{5})}^{3} (n^{3}))}{8 (3^{3} (8 m^{8}))}$

Step 3.2.2.2

Cancel the common factor.

$\frac{8 (4^{3} {(m^{5})}^{3} (n^{3}))}{8 (3^{3} (8 m^{8}))}$

Step 3.2.2.3

Rewrite the expression.

$\frac{4^{3} {(m^{5})}^{3} (n^{3})}{3^{3} (8 m^{8})}$

Step 4

Simplify the numerator.

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Step 4.1

Raise $4$ to the power of $3$ .

$\frac{64 {(m^{5})}^{3} n^{3}}{3^{3} \cdot 8 m^{8}}$

Step 4.2

Multiply the exponents in ${(m^{5})}^{3}$ .

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Step 4.2.1

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

$\frac{64 m^{5 \cdot 3} n^{3}}{3^{3} \cdot 8 m^{8}}$

Step 4.2.2

Multiply $5$ by $3$ .

$\frac{64 m^{15} n^{3}}{3^{3} \cdot 8 m^{8}}$

Step 5

Simplify the denominator.

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Step 5.1

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

$\frac{64 m^{15} n^{3}}{27 \cdot 8 m^{8}}$

Step 5.2

Multiply $27$ by $8$ .

$\frac{64 m^{15} n^{3}}{216 m^{8}}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $64$ and $216$ .

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Step 6.1.1

Factor $8$ out of $64 m^{15} n^{3}$ .

$\frac{8 (8 m^{15} n^{3})}{216 m^{8}}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $8$ out of $216 m^{8}$ .

$\frac{8 (8 m^{15} n^{3})}{8 (27 m^{8})}$

Step 6.1.2.2

Cancel the common factor.

$\frac{8 (8 m^{15} n^{3})}{8 (27 m^{8})}$

Step 6.1.2.3

Rewrite the expression.

$\frac{8 m^{15} n^{3}}{27 m^{8}}$

Step 6.2

Cancel the common factor of $m^{15}$ and $m^{8}$ .

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Step 6.2.1

Factor $m^{8}$ out of $8 m^{15} n^{3}$ .

$\frac{m^{8} (8 m^{7} n^{3})}{27 m^{8}}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $m^{8}$ out of $27 m^{8}$ .

$\frac{m^{8} (8 m^{7} n^{3})}{m^{8} \cdot 27}$

Step 6.2.2.2

Cancel the common factor.

$\frac{m^{8} (8 m^{7} n^{3})}{m^{8} \cdot 27}$

Step 6.2.2.3

Rewrite the expression.

$\frac{8 m^{7} n^{3}}{27}$