Simplify 3/4m^7n^6*m^5n^9-(1/2m^4n^5)^3

$\frac{3}{4} m^{7} n^{6} \cdot m^{5} n^{9} - {(\frac{1}{2} m^{4} n^{5})}^{3}$

Step 1

Simplify each term.

Multiply $m^{7}$ by $m^{5}$ by adding the exponents.

Move $m^{5}$ .

$\frac{3}{4} \cdot (m^{5} m^{7} n^{6}) \cdot n^{9} - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

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

$\frac{3}{4} \cdot (m^{5 + 7} n^{6}) \cdot n^{9} - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Add $5$ and $7$ .

$\frac{3}{4} \cdot (m^{12} n^{6}) \cdot n^{9} - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Multiply $n^{6}$ by $n^{9}$ by adding the exponents.

Move $n^{9}$ .

$\frac{3}{4} \cdot (m^{12} (n^{9} n^{6})) - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

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

$\frac{3}{4} \cdot (m^{12} n^{9 + 6}) - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Add $9$ and $6$ .

$\frac{3}{4} \cdot (m^{12} n^{15}) - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Multiply $\frac{3}{4} (m^{12} n^{15})$ .

Combine $m^{12}$ and $\frac{3}{4}$ .

$\frac{m^{12} \cdot 3}{4} n^{15} - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Combine $\frac{m^{12} \cdot 3}{4}$ and $n^{15}$ .

$\frac{m^{12} \cdot 3 n^{15}}{4} - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Move $3$ to the left of $m^{12}$ .

$\frac{3 \cdot m^{12} n^{15}}{4} - {(\frac{1}{2} \cdot (m^{4} n^{5}))}^{3}$

Multiply $\frac{1}{2} (m^{4} n^{5})$ .

Combine $m^{4}$ and $\frac{1}{2}$ .

$\frac{3 m^{12} n^{15}}{4} - {(\frac{m^{4}}{2} n^{5})}^{3}$

Combine $\frac{m^{4}}{2}$ and $n^{5}$ .

$\frac{3 m^{12} n^{15}}{4} - {(\frac{m^{4} n^{5}}{2})}^{3}$

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

Apply the product rule to $\frac{m^{4} n^{5}}{2}$ .

$\frac{3 m^{12} n^{15}}{4} - \frac{{(m^{4} n^{5})}^{3}}{2^{3}}$

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

$\frac{3 m^{12} n^{15}}{4} - \frac{{(m^{4})}^{3} {(n^{5})}^{3}}{2^{3}}$

Simplify the numerator.

Multiply the exponents in ${(m^{4})}^{3}$ .

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

$\frac{3 m^{12} n^{15}}{4} - \frac{m^{4 \cdot 3} {(n^{5})}^{3}}{2^{3}}$

Multiply $4$ by $3$ .

$\frac{3 m^{12} n^{15}}{4} - \frac{m^{12} {(n^{5})}^{3}}{2^{3}}$

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

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

$\frac{3 m^{12} n^{15}}{4} - \frac{m^{12} n^{5 \cdot 3}}{2^{3}}$

Multiply $5$ by $3$ .

$\frac{3 m^{12} n^{15}}{4} - \frac{m^{12} n^{15}}{2^{3}}$

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

$\frac{3 m^{12} n^{15}}{4} - \frac{m^{12} n^{15}}{8}$

Step 2

To write $\frac{3 m^{12} n^{15}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 m^{12} n^{15}}{4} \cdot \frac{2}{2} - \frac{m^{12} n^{15}}{8}$

Step 3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 m^{12} n^{15}}{4}$ by $\frac{2}{2}$ .

$\frac{3 m^{12} n^{15} \cdot 2}{4 \cdot 2} - \frac{m^{12} n^{15}}{8}$

Multiply $4$ by $2$ .

$\frac{3 m^{12} n^{15} \cdot 2}{8} - \frac{m^{12} n^{15}}{8}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 m^{12} n^{15} \cdot 2 - m^{12} n^{15}}{8}$

Step 5

Simplify the numerator.

Factor $m^{12} n^{15}$ out of $3 m^{12} n^{15} \cdot 2 - m^{12} n^{15}$ .

Factor $m^{12} n^{15}$ out of $3 m^{12} n^{15} \cdot 2$ .

$\frac{m^{12} n^{15} (3 \cdot 2) - m^{12} n^{15}}{8}$

Factor $m^{12} n^{15}$ out of $- m^{12} n^{15}$ .

$\frac{m^{12} n^{15} (3 \cdot 2) + m^{12} n^{15} (- 1)}{8}$

Factor $m^{12} n^{15}$ out of $m^{12} n^{15} (3 \cdot 2) + m^{12} n^{15} (- 1)$ .

$\frac{m^{12} n^{15} (3 \cdot 2 - 1)}{8}$

Multiply $3$ by $2$ .

$\frac{m^{12} n^{15} (6 - 1)}{8}$

Subtract $1$ from $6$ .

$\frac{m^{12} n^{15} \cdot 5}{8}$

Step 6

Move $5$ to the left of $m^{12} n^{15}$ .

$\frac{5 m^{12} n^{15}}{8}$

Simplify 3/4m^7n^6*m^5n^9-(1/2m^4n^5)^3

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