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QuestionSimplify (2x−2y33xy−4)4\left(\frac{2 x^{-2} y^{3}}{3 x y^{-4}}\right)^{4} using positive exponents and evaluate numerical powers.

Studdy Solution

STEP 1

Assumptions1. We have the expression (x−y33xy−4)4\left(\frac{ x^{-} y^{3}}{3 x y^{-4}}\right)^{4}. . We want to simplify this expression and write the answer using only positive exponents.
3. We also want to evaluate any numerical powers.

STEP 2

First, we need to simplify the expression inside the parentheses. We can do this by using the properties of exponents.
2x−2yxy−4=2⋅x−2x⋅yy−4\frac{2 x^{-2} y^{}}{ x y^{-4}} = \frac{2}{} \cdot \frac{x^{-2}}{x} \cdot \frac{y^{}}{y^{-4}}

STEP 3

Next, we simplify the expression further by subtracting the exponents of the same base.
23⋅x−2x⋅y3y−=23⋅x−2−1⋅y3−(−)\frac{2}{3} \cdot \frac{x^{-2}}{x} \cdot \frac{y^{3}}{y^{-}} = \frac{2}{3} \cdot x^{-2-1} \cdot y^{3-(-)}

STEP 4

implify the exponents.
23⋅x−2−1⋅y3−(−4)=23⋅x−3⋅y7\frac{2}{3} \cdot x^{-2-1} \cdot y^{3-(-4)} = \frac{2}{3} \cdot x^{-3} \cdot y^{7}

STEP 5

Now, we have the expression (23x−3y7)4\left(\frac{2}{3} x^{-3} y^{7}\right)^{4}. We can simplify this by applying the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}.
(23x−3y7)4=(23)4⋅(x−3)4⋅(y7)4\left(\frac{2}{3} x^{-3} y^{7}\right)^{4} = \left(\frac{2}{3}\right)^{4} \cdot (x^{-3})^{4} \cdot (y^{7})^{4}

STEP 6

implify the expression.
(23)4⋅(x−3)4⋅(y)4=1681⋅x−12⋅y28\left(\frac{2}{3}\right)^{4} \cdot (x^{-3})^{4} \cdot (y^{})^{4} = \frac{16}{81} \cdot x^{-12} \cdot y^{28}

STEP 7

Finally, we rewrite the expression using only positive exponents. We can do this by using the rule a−n=1ana^{-n} = \frac{1}{a^n}.
1681⋅x−12⋅y28=1681⋅1x12⋅y28\frac{16}{81} \cdot x^{-12} \cdot y^{28} = \frac{16}{81} \cdot \frac{1}{x^{12}} \cdot y^{28}

STEP 8

implify the expression.
1681⋅1x12⋅y28=16y2881x12\frac{16}{81} \cdot \frac{1}{x^{12}} \cdot y^{28} = \frac{16 y^{28}}{81 x^{12}}So, the simplified expression is 16y2881x12\frac{16 y^{28}}{81 x^{12}}.

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