QuestionSimplify $\left(\frac{3 x^{3}}{x^{-2}}\right)^{4} \cdot\left(\frac{y^{2} x^{-4}}{5 x y^{-8}}\right)^{3}$ using positive exponents.

Studdy Solution

STEP 1

Assumptions1. We are given the expression $\left(\frac{3 x^{3} y^{0}}{x^{-}}\right)^{4} \cdot\left(\frac{y^{} x^{-4}}{5 x y^{-8}}\right)^{3}$ . . We need to simplify this expression and write the answer using only positive exponents.
3. We know that any number raised to the power of0 is1.
4. We know that the exponent rules $a^{m} \cdot a^{n} = a^{m+n}$ , $\frac{a^{m}}{a^{n}} = a^{m-n}$ , and $(a^{m})^{n} = a^{mn}$ .

STEP 2

First, simplify the expression inside the first parentheses. $\frac{ x^{} y^{0}}{x^{-2}} = x^{+2} y^{0} = x^{5} y^{0} = x^{5} \cdot1 = x^{5}$

STEP 3

Raise the simplified expression to the power of.
$(3 x^{5})^{} =3^{} (x^{5})^{} =81 x^{20}$

STEP 4

Next, simplify the expression inside the second parentheses.
$\frac{y^{2} x^{-4}}{ x y^{-8}} = \frac{1}{} y^{2+8} x^{-4-1} = \frac{1}{} y^{10} x^{-}$

STEP 5

Raise the simplified expression to the power of3.
$\left(\frac{1}{5} y^{10} x^{-5}\right)^{3} = \left(\frac{1}{5}\right)^{3} (y^{10})^{3} (x^{-5})^{3} = \frac{1}{125} y^{30} x^{-15}$

STEP 6

Finally, multiply the two simplified expressions together.
$81 x^{20} \cdot \frac{1}{125} y^{30} x^{-15} = \frac{81}{125} x^{20-15} y^{30} = \frac{81}{125} x^{5} y^{30}$ The simplified expression is $\frac{81}{125} x^{5} y^{30}$ .

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