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Simplify. Express your answer using exponents.((10x^(9))/(x^(18)))^(-5)

Simplify. Express your answer using exponents. $\newline$ $\left(\frac{10 x^{9}}{x^{18}}\right)^{-5}$

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Evaluate rational exponents

Full solution

Q. Simplify. Express your answer using exponents.

\newline

\left(\frac{10 x^{9}}{x^{18}}\right)^{-5}

Identify base and exponent: Identify the base and the exponent in the expression $((10x^{9})/(x^{18}))^{-5}$ . $\newline$ In $((10x^{9})/(x^{18}))^{-5}$ , the base is $(10x^{9})/(x^{18})$ and the exponent is $-5$ .
Simplify base by dividing powers: Simplify the base $(10x^{9})/(x^{18})$ by dividing the powers of $x$ . Using the quotient rule for exponents, $x^{a}/x^{b} = x^{a-b}$ , we get: $(10x^{9})/(x^{18}) = 10/x^{18-9} = 10/x^{9}$
Apply negative exponent rule: Apply the negative exponent rule to the simplified base. $\newline$ The negative exponent rule states that $(a/b)^{-n} = (b/a)^n$ . Applying this to our expression, we get: $\newline$ $((10/x^{9}))^{-5} = (x^{9}/10)^5$
Expand exponent over fraction: Expand the exponent over the fraction. $\newline$ Using the power of a quotient rule, $(a/b)^n = a^n/b^n$ , we expand the exponent over the numerator and the denominator: $\newline$ $(x^{9}/10)^5 = x^{9*5}/10^5$
Calculate powers of x and $10$ : Calculate the powers of x and $10$ . $\newline$ $x^{9\times 5} = x^{45}$ $\newline$ $10^5 = 100,000$ $\newline$ So, $(x^{9}/10)^5 = x^{45}/100,000$

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