Simplify (12z^-6)/(4(b^-4z^-4)^-2)

$\frac{12 z^{- 6}}{4 {(b^{- 4} z^{- 4})}^{- 2}}$

Step 1

Reduce the expression by cancelling the common factors.

Move $z^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{12}{4 {(b^{- 4} z^{- 4})}^{- 2} z^{6}}$

Move ${(b^{- 4} z^{- 4})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{12 {(b^{- 4} z^{- 4})}^{2}}{4 z^{6}}$

Cancel the common factor of $12$ and $4$ .

Factor $4$ out of $12 {(b^{- 4} z^{- 4})}^{2}$ .

$\frac{4 (3 {(b^{- 4} z^{- 4})}^{2})}{4 z^{6}}$

Cancel the common factors.

Factor $4$ out of $4 z^{6}$ .

$\frac{4 (3 {(b^{- 4} z^{- 4})}^{2})}{4 (z^{6})}$

Cancel the common factor.

$\frac{4 (3 {(b^{- 4} z^{- 4})}^{2})}{4 z^{6}}$

Rewrite the expression.

$\frac{3 {(b^{- 4} z^{- 4})}^{2}}{z^{6}}$

Step 2

Simplify the numerator.

Apply the product rule to $b^{- 4} z^{- 4}$ .

$\frac{3 {(b^{- 4})}^{2} {(z^{- 4})}^{2}}{z^{6}}$

Multiply the exponents in ${(b^{- 4})}^{2}$ .

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

$\frac{3 b^{- 4 \cdot 2} {(z^{- 4})}^{2}}{z^{6}}$

Multiply $- 4$ by $2$ .

$\frac{3 b^{- 8} {(z^{- 4})}^{2}}{z^{6}}$

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

$\frac{3 \frac{1}{b^{8}} {(z^{- 4})}^{2}}{z^{6}}$

Multiply the exponents in ${(z^{- 4})}^{2}$ .

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

$\frac{3 \frac{1}{b^{8}} z^{- 4 \cdot 2}}{z^{6}}$

Multiply $- 4$ by $2$ .

$\frac{3 \frac{1}{b^{8}} z^{- 8}}{z^{6}}$

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

$\frac{3 \frac{1}{b^{8}} \frac{1}{z^{8}}}{z^{6}}$

Combine exponents.

Combine $3$ and $\frac{1}{b^{8}}$ .

$\frac{\frac{3}{b^{8}} \cdot \frac{1}{z^{8}}}{z^{6}}$

Multiply $\frac{3}{b^{8}}$ by $\frac{1}{z^{8}}$ .

$\frac{\frac{3}{b^{8} z^{8}}}{z^{6}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{b^{8} z^{8}} \cdot \frac{1}{z^{6}}$

Step 4

Combine.

$\frac{3 \cdot 1}{b^{8} z^{8} z^{6}}$

Step 5

Multiply $z^{8}$ by $z^{6}$ by adding the exponents.

Move $z^{6}$ .

$\frac{3 \cdot 1}{b^{8} (z^{6} z^{8})}$

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

$\frac{3 \cdot 1}{b^{8} z^{6 + 8}}$

Add $6$ and $8$ .

$\frac{3 \cdot 1}{b^{8} z^{14}}$

Step 6

Multiply $3$ by $1$ .

$\frac{3}{b^{8} z^{14}}$

Simplify (12z^-6)/(4(b^-4z^-4)^-2)

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