Evaluate (8a^(m+3))^3*(1/4a^(1-m))^6

${(8 a^{m + 3})}^{3} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

Step 1

Simplify the expression.

Apply the product rule to $8 a^{m + 3}$ .

$8^{3} {(a^{m + 3})}^{3} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

Raise $8$ to the power of $3$ .

$512 {(a^{m + 3})}^{3} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

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

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

$512 a^{(m + 3) \cdot 3} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

Apply the distributive property.

$512 a^{m \cdot 3 + 3 \cdot 3} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

Move $3$ to the left of $m$ .

$512 a^{3 \cdot m + 3 \cdot 3} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

Multiply $3$ by $3$ .

$512 a^{3 m + 9} \cdot {(\frac{1}{4} a^{1 - m})}^{6}$

Step 2

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

$512 a^{3 m + 9} \cdot {(\frac{a^{1 - m}}{4})}^{6}$

Step 3

Simplify the expression.

Apply the product rule to $\frac{a^{1 - m}}{4}$ .

$512 a^{3 m + 9} \cdot \frac{{(a^{1 - m})}^{6}}{4^{6}}$

Multiply the exponents in ${(a^{1 - m})}^{6}$ .

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

$512 a^{3 m + 9} \cdot \frac{a^{(1 - m) \cdot 6}}{4^{6}}$

Apply the distributive property.

$512 a^{3 m + 9} \cdot \frac{a^{1 \cdot 6 - m \cdot 6}}{4^{6}}$

Multiply $6$ by $1$ .

$512 a^{3 m + 9} \cdot \frac{a^{6 - m \cdot 6}}{4^{6}}$

Multiply $6$ by $- 1$ .

$512 a^{3 m + 9} \cdot \frac{a^{6 - 6 m}}{4^{6}}$

Raise $4$ to the power of $6$ .

$512 a^{3 m + 9} \cdot \frac{a^{6 - 6 m}}{4096}$

Step 4

Cancel the common factor of $512$ .

Factor $512$ out of $512 a^{3 m + 9}$ .

$512 (a^{3 m + 9}) \cdot \frac{a^{6 - 6 m}}{4096}$

Factor $512$ out of $4096$ .

$512 (a^{3 m + 9}) \cdot \frac{a^{6 - 6 m}}{512 (8)}$

Cancel the common factor.

$512 a^{3 m + 9} \cdot \frac{a^{6 - 6 m}}{512 \cdot 8}$

Rewrite the expression.

$a^{3 m + 9} \cdot \frac{a^{6 - 6 m}}{8}$

Step 5

Combine $a^{3 m + 9}$ and $\frac{a^{6 - 6 m}}{8}$ .

$\frac{a^{3 m + 9} a^{6 - 6 m}}{8}$

Step 6

Multiply $a^{3 m + 9}$ by $a^{6 - 6 m}$ by adding the exponents.

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

$\frac{a^{3 m + 9 + 6 - 6 m}}{8}$

Subtract $6 m$ from $3 m$ .

$\frac{a^{- 3 m + 9 + 6}}{8}$

Add $9$ and $6$ .

$\frac{a^{- 3 m + 15}}{8}$

Evaluate (8a^(m+3))^3*(1/4a^(1-m))^6

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