Simplify (y^-4)(100y^8z^10)^(1/2)

$(y^{- 4}) {(100 y^{8} z^{10})}^{\frac{1}{2}}$

Step 1

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

$\frac{1}{y^{4}} {(100 y^{8} z^{10})}^{\frac{1}{2}}$

Step 2

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

Apply the product rule to $100 y^{8} z^{10}$ .

$\frac{1}{y^{4}} ({(100 y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Apply the product rule to $100 y^{8}$ .

$\frac{1}{y^{4}} (100^{\frac{1}{2}} {(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Step 3

Simplify the expression.

Rewrite using the commutative property of multiplication.

$100^{\frac{1}{2}} \frac{1}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Rewrite $100$ as $10^{2}$ .

${(10^{2})}^{\frac{1}{2}} \frac{1}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

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

$10^{2 (\frac{1}{2})} \frac{1}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Step 4

Cancel the common factor of $2$ .

Cancel the common factor.

$10^{2 (\frac{1}{2})} \frac{1}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Rewrite the expression.

$10^{1} \frac{1}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Step 5

Evaluate the exponent.

$10 \frac{1}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Step 6

Combine $10$ and $\frac{1}{y^{4}}$ .

$\frac{10}{y^{4}} ({(y^{8})}^{\frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Step 7

Multiply the exponents in ${(y^{8})}^{\frac{1}{2}}$ .

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

$\frac{10}{y^{4}} (y^{8 (\frac{1}{2})} {(z^{10})}^{\frac{1}{2}})$

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

$\frac{10}{y^{4}} (y^{2 (4) \frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Cancel the common factor.

$\frac{10}{y^{4}} (y^{2 \cdot 4 \frac{1}{2}} {(z^{10})}^{\frac{1}{2}})$

Rewrite the expression.

$\frac{10}{y^{4}} (y^{4} {(z^{10})}^{\frac{1}{2}})$

Step 8

Cancel the common factor of $y^{4}$ .

Factor $y^{4}$ out of $y^{4} {(z^{10})}^{\frac{1}{2}}$ .

$\frac{10}{y^{4}} (y^{4} ({(z^{10})}^{\frac{1}{2}}))$

Cancel the common factor.

$\frac{10}{y^{4}} (y^{4} {(z^{10})}^{\frac{1}{2}})$

Rewrite the expression.

$10 {(z^{10})}^{\frac{1}{2}}$

Step 9

Multiply the exponents in ${(z^{10})}^{\frac{1}{2}}$ .

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

$10 z^{10 (\frac{1}{2})}$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$10 z^{2 (5) \frac{1}{2}}$

Cancel the common factor.

$10 z^{2 \cdot 5 \frac{1}{2}}$

Rewrite the expression.

$10 z^{5}$

Simplify (y^-4)(100y^8z^10)^(1/2)

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