301 Moved Permanently - Free Printable

Problem Analysis:

The worksheet involves converting expressions with negative exponents into their equivalent forms. The key properties of exponents that we will use are:

1. Negative Exponent Rule:
\[
a^{-n} = \frac{1}{a^n}
\]
This means that a negative exponent can be converted to a positive exponent by taking the reciprocal of the base.

2. Reciprocal of a Fraction:
If the base is a fraction, say \(\left(\frac{p}{q}\right)^{-n}\), then:
\[
\left(\frac{p}{q}\right)^{-n} = \left(\frac{q}{p}\right)^n
\]

Let's solve each problem step by step.

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Question 25:

Expression of \(\left(\frac{1}{4}\right)^3\) as a rational number with a negative exponent is ______.

#### Solution:
We need to express \(\left(\frac{1}{4}\right)^3\) in terms of a negative exponent. Using the negative exponent rule:
\[
\left(\frac{1}{4}\right)^3 = 4^{-3}
\]
Thus, the correct answer is:
\[
\boxed{A}
\]

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Question 26:

Expression of \(3^5\) as a rational number with a negative exponent is ______.

#### Solution:
We need to express \(3^5\) in terms of a negative exponent. Using the negative exponent rule:
\[
3^5 = \left(\frac{1}{3}\right)^{-5}
\]
Thus, the correct answer is:
\[
\boxed{B}
\]

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Question 27:

Expression of \(\left(\frac{3}{5}\right)^4\) as a rational number with a negative exponent is ______.

#### Solution:
We need to express \(\left(\frac{3}{5}\right)^4\) in terms of a negative exponent. Using the reciprocal property for fractions:
\[
\left(\frac{3}{5}\right)^4 = \left(\frac{5}{3}\right)^{-4}
\]
Thus, the correct answer is:
\[
\boxed{D}
\]

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Question 28:

Expression of \(\left\{\left(\frac{3}{2}\right)^4\right\}^{-3}\) as a rational number with a negative exponent is ______.

#### Solution:
We start with the given expression:
\[
\left\{\left(\frac{3}{2}\right)^4\right\}^{-3}
\]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
\left\{\left(\frac{3}{2}\right)^4\right\}^{-3} = \left(\frac{3}{2}\right)^{4 \cdot (-3)} = \left(\frac{3}{2}\right)^{-12}
\]
Thus, the correct answer is:
\[
\boxed{C}
\]

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Question 29:

Expression of \(\left\{\left(\frac{7}{3}\right)^4\right\}^{-3}\) as a rational number with a negative exponent is ______.

#### Solution:
We start with the given expression:
\[
\left\{\left(\frac{7}{3}\right)^4\right\}^{-3}
\]
Using the property of exponents \((a^m)^n = a^{m \cdot n}\):
\[
\left\{\left(\frac{7}{3}\right)^4\right\}^{-3} = \left(\frac{7}{3}\right)^{4 \cdot (-3)} = \left(\frac{7}{3}\right)^{-12}
\]
Thus, the correct answer is:
\[
\boxed{D}
\]

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Question 30:

Expression of \(\left(\frac{3}{4}\right)^{-2}\) as a rational number with a positive exponent is ______.

#### Solution:
We need to convert \(\left(\frac{3}{4}\right)^{-2}\) into a positive exponent. Using the negative exponent rule:
\[
\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2
\]
Thus, the correct answer is:
\[
\boxed{C}
\]

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Final Answers:

\[
\boxed{A, B, D, C, D, C}
\]