Simplify (2j^2k^4)^-5(k^-1j^7)^6

${(2 j^{2} k^{4})}^{- 5} {(k^{- 1} j^{7})}^{6}$

Step 1

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

$\frac{1}{{(2 j^{2} k^{4})}^{5}} {(k^{- 1} j^{7})}^{6}$

Step 2

Simplify the denominator.

Apply the product rule to $2 j^{2} k^{4}$ .

$\frac{1}{{(2 j^{2})}^{5} {(k^{4})}^{5}} {(k^{- 1} j^{7})}^{6}$

Apply the product rule to $2 j^{2}$ .

$\frac{1}{2^{5} {(j^{2})}^{5} {(k^{4})}^{5}} {(k^{- 1} j^{7})}^{6}$

Raise $2$ to the power of $5$ .

$\frac{1}{32 {(j^{2})}^{5} {(k^{4})}^{5}} {(k^{- 1} j^{7})}^{6}$

Multiply the exponents in ${(j^{2})}^{5}$ .

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

$\frac{1}{32 j^{2 \cdot 5} {(k^{4})}^{5}} {(k^{- 1} j^{7})}^{6}$

Multiply $2$ by $5$ .

$\frac{1}{32 j^{10} {(k^{4})}^{5}} {(k^{- 1} j^{7})}^{6}$

Multiply the exponents in ${(k^{4})}^{5}$ .

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

$\frac{1}{32 j^{10} k^{4 \cdot 5}} {(k^{- 1} j^{7})}^{6}$

Multiply $4$ by $5$ .

$\frac{1}{32 j^{10} k^{20}} {(k^{- 1} j^{7})}^{6}$

Step 3

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

$\frac{1}{32 j^{10} k^{20}} {(\frac{1}{k} j^{7})}^{6}$

Step 4

Combine $\frac{1}{k}$ and $j^{7}$ .

$\frac{1}{32 j^{10} k^{20}} {(\frac{j^{7}}{k})}^{6}$

Step 5

Apply the product rule to $\frac{j^{7}}{k}$ .

$\frac{1}{32 j^{10} k^{20}} \cdot \frac{{(j^{7})}^{6}}{k^{6}}$

Step 6

Combine.

$\frac{1 {(j^{7})}^{6}}{32 j^{10} k^{20} k^{6}}$

Step 7

Multiply $k^{20}$ by $k^{6}$ by adding the exponents.

Move $k^{6}$ .

$\frac{1 {(j^{7})}^{6}}{32 j^{10} (k^{6} k^{20})}$

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

$\frac{1 {(j^{7})}^{6}}{32 j^{10} k^{6 + 20}}$

Add $6$ and $20$ .

$\frac{1 {(j^{7})}^{6}}{32 j^{10} k^{26}}$

Step 8

Simplify the expression.

Multiply ${(j^{7})}^{6}$ by $1$ .

$\frac{{(j^{7})}^{6}}{32 j^{10} k^{26}}$

Multiply the exponents in ${(j^{7})}^{6}$ .

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

$\frac{j^{7 \cdot 6}}{32 j^{10} k^{26}}$

Multiply $7$ by $6$ .

$\frac{j^{42}}{32 j^{10} k^{26}}$

Step 9

Cancel the common factor of $j^{42}$ and $j^{10}$ .

Factor $j^{10}$ out of $j^{42}$ .

$\frac{j^{10} j^{32}}{32 j^{10} k^{26}}$

Cancel the common factors.

Factor $j^{10}$ out of $32 j^{10} k^{26}$ .

$\frac{j^{10} j^{32}}{j^{10} (32 k^{26})}$

Cancel the common factor.

$\frac{j^{10} j^{32}}{j^{10} (32 k^{26})}$

Rewrite the expression.

$\frac{j^{32}}{32 k^{26}}$

Simplify (2j^2k^4)^-5(k^-1j^7)^6

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