Simplify ((2j^-4y^5)^3)/((3j^6y^3)^3)

$\frac{{(2 j^{- 4} y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

Step 1

Simplify the numerator.

Apply the product rule to $2 j^{- 4} y^{5}$ .

$\frac{{(2 j^{- 4})}^{3} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

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

$\frac{{(2 \frac{1}{j^{4}})}^{3} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

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

$\frac{{(\frac{2}{j^{4}})}^{3} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

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

$\frac{\frac{2^{3}}{{(j^{4})}^{3}} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

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

$\frac{\frac{8}{{(j^{4})}^{3}} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

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

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

$\frac{\frac{8}{j^{4 \cdot 3}} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

Multiply $4$ by $3$ .

$\frac{\frac{8}{j^{12}} {(y^{5})}^{3}}{{(3 j^{6} y^{3})}^{3}}$

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

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

$\frac{\frac{8}{j^{12}} y^{5 \cdot 3}}{{(3 j^{6} y^{3})}^{3}}$

Multiply $5$ by $3$ .

$\frac{\frac{8}{j^{12}} y^{15}}{{(3 j^{6} y^{3})}^{3}}$

Step 2

Simplify the denominator.

Apply the product rule to $3 j^{6} y^{3}$ .

$\frac{\frac{8}{j^{12}} y^{15}}{{(3 j^{6})}^{3} {(y^{3})}^{3}}$

Apply the product rule to $3 j^{6}$ .

$\frac{\frac{8}{j^{12}} y^{15}}{3^{3} {(j^{6})}^{3} {(y^{3})}^{3}}$

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

$\frac{\frac{8}{j^{12}} y^{15}}{27 {(j^{6})}^{3} {(y^{3})}^{3}}$

Multiply the exponents in ${(j^{6})}^{3}$ .

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

$\frac{\frac{8}{j^{12}} y^{15}}{27 j^{6 \cdot 3} {(y^{3})}^{3}}$

Multiply $6$ by $3$ .

$\frac{\frac{8}{j^{12}} y^{15}}{27 j^{18} {(y^{3})}^{3}}$

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

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

$\frac{\frac{8}{j^{12}} y^{15}}{27 j^{18} y^{3 \cdot 3}}$

Multiply $3$ by $3$ .

$\frac{\frac{8}{j^{12}} y^{15}}{27 j^{18} y^{9}}$

Step 3

Combine $\frac{8}{j^{12}}$ and $y^{15}$ .

$\frac{\frac{8 y^{15}}{j^{12}}}{27 j^{18} y^{9}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{8 y^{15}}{j^{12}} \cdot \frac{1}{27 j^{18} y^{9}}$

Step 5

Combine.

$\frac{8 y^{15} \cdot 1}{j^{12} (27 j^{18} y^{9})}$

Step 6

Multiply $j^{12}$ by $j^{18}$ by adding the exponents.

Move $j^{18}$ .

$\frac{8 y^{15} \cdot 1}{j^{18} j^{12} (27 y^{9})}$

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

$\frac{8 y^{15} \cdot 1}{j^{18 + 12} (27 y^{9})}$

Add $18$ and $12$ .

$\frac{8 y^{15} \cdot 1}{j^{30} (27 y^{9})}$

Step 7

Multiply $8 y^{15}$ by $1$ .

$\frac{8 y^{15}}{j^{30} (27 y^{9})}$

Step 8

Cancel the common factor of $y^{15}$ and $y^{9}$ .

Factor $y^{9}$ out of $8 y^{15}$ .

$\frac{y^{9} (8 y^{6})}{j^{30} (27 y^{9})}$

Cancel the common factors.

Factor $y^{9}$ out of $j^{30} (27 y^{9})$ .

$\frac{y^{9} (8 y^{6})}{y^{9} (j^{30} \cdot (27))}$

Cancel the common factor.

$\frac{y^{9} (8 y^{6})}{y^{9} (j^{30} \cdot (27))}$

Rewrite the expression.

$\frac{8 y^{6}}{j^{30} \cdot (27)}$

Step 9

Move $27$ to the left of $j^{30}$ .

$\frac{8 y^{6}}{27 j^{30}}$

Simplify ((2j^-4y^5)^3)/((3j^6y^3)^3)

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