Simplify (-1/2xy^6z^8)^2(1/2x^8y^6z)^2

${(- \frac{1}{2} x y^{6} z^{8})}^{2} {(\frac{1}{2} x^{8} y^{6} z)}^{2}$

Step 1

Multiply $- \frac{1}{2} (x y^{6} z^{8})$ .

Combine $x$ and $\frac{1}{2}$ .

${(- \frac{x}{2} (y^{6} z^{8}))}^{2} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Combine $y^{6}$ and $\frac{x}{2}$ .

${(- \frac{y^{6} x}{2} z^{8})}^{2} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Combine $z^{8}$ and $\frac{y^{6} x}{2}$ .

${(- \frac{z^{8} (y^{6} x)}{2})}^{2} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Step 2

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

Apply the product rule to $- \frac{z^{8} y^{6} x}{2}$ .

${(- 1)}^{2} {(\frac{z^{8} y^{6} x}{2})}^{2} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Apply the product rule to $\frac{z^{8} y^{6} x}{2}$ .

${(- 1)}^{2} \frac{{(z^{8} y^{6} x)}^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Apply the product rule to $z^{8} y^{6} x$ .

${(- 1)}^{2} \frac{{(z^{8} y^{6})}^{2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Apply the product rule to $z^{8} y^{6}$ .

${(- 1)}^{2} \frac{{(z^{8})}^{2} {(y^{6})}^{2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Step 3

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 \frac{{(z^{8})}^{2} {(y^{6})}^{2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Multiply $\frac{{(z^{8})}^{2} {(y^{6})}^{2} x^{2}}{2^{2}}$ by $1$ .

$\frac{{(z^{8})}^{2} {(y^{6})}^{2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Step 4

Simplify the numerator.

Multiply the exponents in ${(z^{8})}^{2}$ .

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

$\frac{z^{8 \cdot 2} {(y^{6})}^{2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Multiply $8$ by $2$ .

$\frac{z^{16} {(y^{6})}^{2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Multiply the exponents in ${(y^{6})}^{2}$ .

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

$\frac{z^{16} y^{6 \cdot 2} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Multiply $6$ by $2$ .

$\frac{z^{16} y^{12} x^{2}}{2^{2}} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Step 5

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

$\frac{z^{16} y^{12} x^{2}}{4} {(\frac{1}{2} \cdot (x^{8} y^{6} z))}^{2}$

Step 6

Multiply $\frac{1}{2} (x^{8} y^{6} z)$ .

Combine $x^{8}$ and $\frac{1}{2}$ .

$\frac{z^{16} y^{12} x^{2}}{4} {(\frac{x^{8}}{2} (y^{6} z))}^{2}$

Combine $y^{6}$ and $\frac{x^{8}}{2}$ .

$\frac{z^{16} y^{12} x^{2}}{4} {(\frac{y^{6} x^{8}}{2} z)}^{2}$

Combine $\frac{y^{6} x^{8}}{2}$ and $z$ .

$\frac{z^{16} y^{12} x^{2}}{4} {(\frac{y^{6} x^{8} z}{2})}^{2}$

Step 7

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

Apply the product rule to $\frac{y^{6} x^{8} z}{2}$ .

$\frac{z^{16} y^{12} x^{2}}{4} \cdot \frac{{(y^{6} x^{8} z)}^{2}}{2^{2}}$

Apply the product rule to $y^{6} x^{8} z$ .

$\frac{z^{16} y^{12} x^{2}}{4} \cdot \frac{{(y^{6} x^{8})}^{2} z^{2}}{2^{2}}$

Apply the product rule to $y^{6} x^{8}$ .

$\frac{z^{16} y^{12} x^{2}}{4} \cdot \frac{{(y^{6})}^{2} {(x^{8})}^{2} z^{2}}{2^{2}}$

Step 8

Combine.

$\frac{z^{16} y^{12} x^{2} ({(y^{6})}^{2} {(x^{8})}^{2} z^{2})}{4 \cdot 2^{2}}$

Step 9

Multiply $z^{16}$ by $z^{2}$ by adding the exponents.

Move $z^{2}$ .

$\frac{z^{2} z^{16} y^{12} x^{2} ({(y^{6})}^{2} {(x^{8})}^{2})}{4 \cdot 2^{2}}$

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

$\frac{z^{2 + 16} y^{12} x^{2} ({(y^{6})}^{2} {(x^{8})}^{2})}{4 \cdot 2^{2}}$

Add $2$ and $16$ .

$\frac{z^{18} y^{12} x^{2} ({(y^{6})}^{2} {(x^{8})}^{2})}{4 \cdot 2^{2}}$

Step 10

Simplify the numerator.

Rewrite.

$\frac{z^{16} y^{12} x^{2} ({(y^{6})}^{2} {(x^{8})}^{2} z^{2})}{4 \cdot 2^{2}}$

Remove unnecessary parentheses.

$\frac{z^{18} y^{12} x^{2} {(y^{6})}^{2} {(x^{8})}^{2}}{4 \cdot 2^{2}}$

Multiply the exponents in ${(y^{6})}^{2}$ .

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

$\frac{z^{18} y^{12} x^{2} y^{6 \cdot 2} {(x^{8})}^{2}}{4 \cdot 2^{2}}$

Multiply $6$ by $2$ .

$\frac{z^{18} y^{12} x^{2} y^{12} {(x^{8})}^{2}}{4 \cdot 2^{2}}$

Multiply the exponents in ${(x^{8})}^{2}$ .

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

$\frac{z^{18} y^{12} x^{2} y^{12} x^{8 \cdot 2}}{4 \cdot 2^{2}}$

Multiply $8$ by $2$ .

$\frac{z^{18} y^{12} x^{2} y^{12} x^{16}}{4 \cdot 2^{2}}$

Combine exponents.

Multiply $y^{12}$ by $y^{12}$ by adding the exponents.

Move $y^{12}$ .

$\frac{z^{18} (y^{12} y^{12}) x^{2} x^{16}}{4 \cdot 2^{2}}$

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

$\frac{z^{18} y^{12 + 12} x^{2} x^{16}}{4 \cdot 2^{2}}$

Add $12$ and $12$ .

$\frac{z^{18} y^{24} x^{2} x^{16}}{4 \cdot 2^{2}}$

Multiply $x^{2}$ by $x^{16}$ by adding the exponents.

Move $x^{16}$ .

$\frac{z^{18} y^{24} (x^{16} x^{2})}{4 \cdot 2^{2}}$

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

$\frac{z^{18} y^{24} x^{16 + 2}}{4 \cdot 2^{2}}$

Add $16$ and $2$ .

$\frac{z^{18} y^{24} x^{18}}{4 \cdot 2^{2}}$

Step 11

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$\frac{z^{18} y^{24} x^{18}}{2^{2} \cdot 2^{2}}$

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

$\frac{z^{18} y^{24} x^{18}}{2^{2 + 2}}$

Add $2$ and $2$ .

$\frac{z^{18} y^{24} x^{18}}{2^{4}}$

Step 12

Raise $2$ to the power of $4$ .

$\frac{z^{18} y^{24} x^{18}}{16}$

Simplify (-1/2xy^6z^8)^2(1/2x^8y^6z)^2

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