Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\frac{4}{b^{2} c^{4}}
\]
Solution
To simplify the expression:
\[
\left(\frac{2 b^{3} c^{2}}{4 b^{2}}\right)^{-2}
\]
Let's break it down step by step.
### Step 1: Simplify the Fraction Inside the Parentheses
First, simplify the expression inside the parentheses:
\[
\frac{2 b^{3} c^{2}}{4 b^{2}}
\]
**Simplify the constants:**
\[
\frac{2}{4} = \frac{1}{2}
\]
**Simplify the variables using the laws of exponents (\( \frac{a^m}{a^n} = a^{m-n} \)):**
\[
\frac{b^{3}}{b^{2}} = b^{3-2} = b^{1} = b
\]
\[
c^{2} \quad (\text{remains as is since there's no } c \text{ in the denominator})
\]
Putting it all together:
\[
\frac{2 b^{3} c^{2}}{4 b^{2}} = \frac{1}{2} \cdot b \cdot c^{2} = \frac{b c^{2}}{2}
\]
### Step 2: Apply the Negative Exponent
Now, apply the exponent \(-2\) to the simplified expression:
\[
\left(\frac{b c^{2}}{2}\right)^{-2}
\]
Recall that \( (a)^{-n} = \frac{1}{a^{n}} \). So,
\[
\left(\frac{b c^{2}}{2}\right)^{-2} = \left(\frac{2}{b c^{2}}\right)^{2}
\]
### Step 3: Square the Expression
Square both the numerator and the denominator:
\[
\left(\frac{2}{b c^{2}}\right)^{2} = \frac{2^{2}}{(b c^{2})^{2}} = \frac{4}{b^{2} c^{4}}
\]
### Final Answer
\[
\boxed{\dfrac{4}{b^{2} c^{4}}}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To simplify the expression \( \left(\frac{2 b^{3} c^{2}}{4 b^{2}}\right)^{-2} \), we first simplify the fraction inside the parentheses. Dividing the numerator and the denominator gives us \( \frac{b^{3} c^{2}}{2 b^{2}} \). This simplifies further to \( \frac{c^{2}}{2} b^{3-2} = \frac{c^{2}}{2} b^{1} = \frac{c^{2} b}{2} \). Next, we take the entire expression to the power of -2: \[ \left( \frac{c^{2} b}{2} \right)^{-2} = \left( 2 \right)^{2} \cdot (c^{2})^{-2} \cdot (b)^{-2} = \frac{4}{c^{4} b^{2}}. \] So, the simplified expression is: \[ \frac{4}{c^{4} b^{2}}. \]
