Question

im2 midterm study guide #3
test overview 1
question 3
rewrite the expression \\(\sqrt5{\frac{a^3 \cdot b^7}{a^2 \cdot b^{-3}}}\\) to exponential form.
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Explanation:

Step1: Simplify the fraction inside the root

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

Step2: Convert the radical to exponential form

Recall that \( \sqrt[n]{x}=x^{\frac{1}{n}} \). Here, \( n = 5 \) and \( x=a\cdot b^{10} \), so \( \sqrt[5]{a\cdot b^{10}}=(a\cdot b^{10})^{\frac{1}{5}} \).

Step3: Apply the power - of - a - product rule

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

Step4: Simplify the exponents of \( b \)

Using the power - of - a - power rule \( (x^m)^n=x^{m\times n} \), for \( (b^{10})^{\frac{1}{5}} \), we get \( b^{10\times\frac{1}{5}}=b^{2} \).

Answer:

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