Question

im2 midterm study guide #3
test overview 12
question 3
rewrite the expression \\(\sqrt5{dfrac{a^3 \cdot b^7}{a^2 \cdot b^{-3}}}\\) to exponential form.
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Explanation:

Step1: Simplify numerator and denominator

First, simplify the fraction inside the fifth root using the quotient rule for exponents \( \frac{x^m}{x^n}=x^{m - n} \). For the \( a \)-terms: \( a^{3-2}=a^1 \). For the \( b \)-terms: \( b^{7-(-3)} = b^{7 + 3}=b^{10} \). So the expression inside the root becomes \( \frac{a^3b^7}{a^2b^{-3}}=a^{3 - 2}b^{7-(-3)}=ab^{10} \).

Step2: Convert the fifth root to an exponent

Recall that \( \sqrt[n]{x}=x^{\frac{1}{n}} \). So \( \sqrt[5]{ab^{10}}=(ab^{10})^{\frac{1}{5}} \).

Step3: Apply the power of a product rule

Using \( (xy)^n=x^n y^n \), we get \( a^{\frac{1}{5}}(b^{10})^{\frac{1}{5}} \).

Step4: Simplify the exponents

For the \( b \)-term, use \( (x^m)^n=x^{mn} \), so \( (b^{10})^{\frac{1}{5}}=b^{10\times\frac{1}{5}} = b^2 \). The \( a \)-term remains \( a^{\frac{1}{5}} \). So the expression is \( a^{\frac{1}{5}}b^2 \).

Answer:

\( a^{\frac{1}{5}}b^{2} \)