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QuestionSimplify the expression with positive exponents: (5x39z−4)−2=\left(\frac{5 x^{3}}{9 z^{-4}}\right)^{-2}=

Studdy Solution

STEP 1

Assumptions1. The expression given is (5x39z−4)−\left(\frac{5 x^{3}}{9 z^{-4}}\right)^{-} . We need to simplify this expression using only positive exponents3. We can use the rule of exponents a−n=1ana^{-n} = \frac{1}{a^n}

STEP 2

We can apply the power of a quotient rule which states (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n}.
\left(\frac{5 x^{}}{9 z^{-4}}\right)^{-2} = \frac{(5x^)^{-2}}{(9z^{-4})^{-2}}

STEP 3

Now, we can use the rule of exponents a−n=1ana^{-n} = \frac{1}{a^n} to rewrite the expression with positive exponents.
(5x3)−2(9z−)−2=1(5x3)21(9z−)2\frac{(5x^3)^{-2}}{(9z^{-})^{-2}} = \frac{\frac{1}{(5x^3)^2}}{\frac{1}{(9z^{-})^2}}

STEP 4

We can simplify the expression by multiplying the numerator and denominator by (x3)2(9z−4)2(x^3)^2(9z^{-4})^2.
1(x3)21(9z−4)2=(9z−4)2(x3)2\frac{\frac{1}{(x^3)^2}}{\frac{1}{(9z^{-4})^2}} = \frac{(9z^{-4})^2}{(x^3)^2}

STEP 5

Now, we can apply the power of a product rule which states (ab)n=anbn(ab)^n = a^n b^n.
(9z−4)2(5x3)2=92(z−4)252(x3)2\frac{(9z^{-4})^2}{(5x^3)^2} = \frac{9^2 (z^{-4})^2}{5^2 (x^3)^2}

STEP 6

We can simplify the expression by evaluating the powers where appropriate.
92(z−4)252(x3)2=81z−825x6\frac{9^2 (z^{-4})^2}{5^2 (x^3)^2} = \frac{81 z^{-8}}{25 x^6}

STEP 7

Finally, we can use the rule of exponents a−n=1ana^{-n} = \frac{1}{a^n} to rewrite the expression with positive exponents.
\frac{81 z^{-}}{25 x^6} = \frac{81}{25 x^6 z^} So, the simplified expression is \frac{81}{25 x^6 z^}.

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