Question

question 2
express the following fraction in simplest form, only using positive exponents.
\frac{10r^2}{(5r^4)^3}
answer attempt 1 out of 2
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Explanation:

Step1: Simplify the denominator's power

First, we simplify \((5r^4)^3\) using the power - of - a - product rule \((ab)^n=a^n\times b^n\). So \((5r^4)^3 = 5^3\times(r^4)^3\). We know that \(5^3=125\) and \((r^4)^3=r^{4\times3}=r^{12}\) (by the power - of - a - power rule \((a^m)^n=a^{m\times n}\)). So the denominator becomes \(125r^{12}\), and our fraction is now \(\frac{10r^{2}}{125r^{12}}\).

Step2: Simplify the coefficient and the variable part separately

For the coefficient part: We find the greatest common divisor (GCD) of 10 and 125. The factors of 10 are 1, 2, 5, 10 and the factors of 125 are 1, 5, 25, 125. The GCD of 10 and 125 is 5. We divide both the numerator and the denominator by 5. \(\frac{10\div5}{125\div5}=\frac{2}{25}\).

For the variable part: We use the quotient rule for exponents \(a^m\div a^n=a^{m - n}\) (where \(a
eq0\) and \(m\geq n\) for positive exponents). Here, for the variable \(r\), we have \(r^{2}\div r^{12}=r^{2-12}=r^{- 10}\). But we want only positive exponents, and we know that \(a^{-n}=\frac{1}{a^{n}}\), so \(r^{-10}=\frac{1}{r^{10}}\).

Step3: Combine the coefficient and the variable part

Now we multiply the simplified coefficient and the simplified variable part. \(\frac{2}{25}\times\frac{1}{r^{10}}=\frac{2}{25r^{10}}\)

Answer:

\(\frac{2}{25r^{10}}\)