Question

im2 midterm study guide #3
test overview 5
question 11
what is the simplified form of this expression? $sqrt3{\frac{y^4}{y^{10}}}$
answer
$\frac{1}{y^2}$ $\frac{1}{y^3}$
$y^2$ $y^3$
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Explanation:

Step1: Simplify the fraction inside the cube root

First, we simplify the fraction \(\frac{y^{4}}{y^{10}}\) using the exponent rule \( \frac{a^{m}}{a^{n}} = a^{m - n} \). So, \( y^{4-10}=y^{-6} \). Now the expression becomes \( \sqrt[3]{y^{-6}} \).

Step2: Apply the cube root exponent rule

The cube root of a number \( a^{m} \) can be written as \( a^{\frac{m}{3}} \). So for \( \sqrt[3]{y^{-6}} \), we have \( y^{\frac{-6}{3}} \).

Step3: Simplify the exponent

Simplify \( \frac{-6}{3}=-2 \), so the expression is \( y^{-2} \). And using the rule \( a^{-n}=\frac{1}{a^{n}} \), \( y^{-2}=\frac{1}{y^{2}} \).

Answer:

\(\frac{1}{y^{2}}\) (corresponding to the option \(\boldsymbol{\frac{1}{y^{2}}}\))