Question
Simplify the following expression and express the final answer with positive exponents: \[ (4 c)^{-3} \] \( \sqrt[3]{4 c} \) \( \frac{1}{(4 c)^{3}} \) \( -64 c^{3} \) \( \frac{1}{\sqrt[3]{4 c}} \) \( \frac{1}{64 c^{3}} \)
Ask by Joseph Vaughn. in Canada
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{1}{64 c^{3}} \).
Solution
We begin with the expression
\[
(4c)^{-3}.
\]
A negative exponent means we take the reciprocal of the base raised to the corresponding positive exponent. Therefore,
\[
(4c)^{-3} = \frac{1}{(4c)^3}.
\]
Next, we simplify the denominator by applying the exponent to both factors:
\[
(4c)^3 = 4^3 \cdot c^3 = 64 c^3.
\]
Substituting this back into the expression, we get:
\[
\frac{1}{(4c)^3} = \frac{1}{64 c^3}.
\]
Thus, the simplified expression with positive exponents is
\[
\frac{1}{64 c^3}.
\]
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Simplify this solution Bonus Knowledge
To simplify \((4c)^{-3}\), you can apply the property of negative exponents which states that \(a^{-n} = \frac{1}{a^{n}}\). This gives us: \[ (4c)^{-3} = \frac{1}{(4c)^{3}} \] Now, let's calculate \((4c)^{3}\): \[ (4c)^{3} = 4^{3} \cdot c^{3} = 64c^{3} \] Therefore, we have: \[ (4c)^{-3} = \frac{1}{64c^{3}} \] So the final answer, expressed with positive exponents, is: \[ \frac{1}{64c^{3}} \]
