301 Moved Permanently - Free Printable

Let's solve each question step by step, using the rules of exponents and rational numbers.

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Key Concepts:

1. $ a^{-n} = \frac{1}{a^n} $
2. $ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $
3. $ \left(\left(\frac{a}{b}\right)^m\right)^n = \left(\frac{a}{b}\right)^{m \cdot n} $
4. Negative exponents mean reciprocal; positive exponents mean original form.

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Question 25:

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

We have:
$$
\left(\frac{1}{4}\right)^3 = \frac{1^3}{4^3} = \frac{1}{64}
$$

Now, we want to express this using a negative exponent.

Note:
$$
\frac{1}{64} = \frac{1}{4^3} = 4^{-3}
$$

So, the correct expression with a negative exponent is:
$$
\boxed{4^{-3}}
$$

✔ Answer: (A) $4^{-3}$

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Question 26:

Expression of $3^5$ as a rational number with a negative exponent is _______.

We know:
$$
3^5 = \frac{1}{3^{-5}} \Rightarrow 3^5 = \left(\frac{1}{3}\right)^{-5}
$$

Because:
$$
\left(\frac{1}{3}\right)^{-5} = \frac{1}{(1/3)^5} = \frac{1}{1/243} = 243 = 3^5
$$

So, $3^5 = \left(\frac{1}{3}\right)^{-5}$

✔ Answer: (B) $\left(\frac{1}{3}\right)^{-5}$

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Question 27:

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

We know:
$$
\left(\frac{3}{5}\right)^4 = \left(\frac{5}{3}\right)^{-4}
$$

Why? Because:
$$
\left(\frac{a}{b}\right)^n = \left(\frac{b}{a}\right)^{-n}
\Rightarrow \left(\frac{3}{5}\right)^4 = \left(\frac{5}{3}\right)^{-4}
$$

But look at the options:

(A) $-\left(\frac{5}{3}\right)^{-4}$ → wrong sign
(B) $\left(\frac{25}{9}\right)^{-4}$ → not equivalent
(C) $\left(\frac{9}{25}\right)^{-4}$ → no
(D) $\left(\frac{5}{3}\right)^{-4}$ → ✔ Correct!

Wait — let's verify:

Is $\left(\frac{3}{5}\right)^4 = \left(\frac{5}{3}\right)^{-4}$?

Yes! Because:
$$
\left(\frac{5}{3}\right)^{-4} = \frac{1}{(5/3)^4} = \frac{1}{625/81} = \frac{81}{625} = \left(\frac{3}{5}\right)^4
$$

So yes, it matches.

✔ Answer: (D) $\left(\frac{5}{3}\right)^{-4}$

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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 _______.

First simplify:
$$
\left(\left(\frac{3}{2}\right)^4\right)^{-3} = \left(\frac{3}{2}\right)^{-12}
$$

Now, convert this to a rational number with negative exponent, but we already have a negative exponent.

But we can write:
$$
\left(\frac{3}{2}\right)^{-12} = \left(\frac{2}{3}\right)^{12}
$$

But the question asks for an expression with a negative exponent.

So which option has a negative exponent?

Look at options:

(A) $-\left(\frac{3}{2}\right)^{-12}$ → extra negative sign → incorrect
(B) $\left(\frac{2}{3}\right)^{-12}$ → this is equal to $\left(\frac{3}{2}\right)^{12}$ → not same
(C) $\left(\frac{3}{2}\right)^{-12}$ → ✔ This is exactly what we got
(D) $-\left(\frac{2}{3}\right)^{-12}$ → negative sign → incorrect

So, our answer is:
$$
\left(\frac{3}{2}\right)^{-12}
$$

✔ Answer: (C) $\left(\frac{3}{2}\right)^{-12}$

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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 _______.

Simplify:
$$
\left(\left(\frac{7}{3}\right)^4\right)^{-3} = \left(\frac{7}{3}\right)^{-12}
$$

Now, convert to a rational number with negative exponent.

We can write:
$$
\left(\frac{7}{3}\right)^{-12} = \left(\frac{3}{7}\right)^{12}
$$

But again, the question wants a negative exponent.

So, keep it as:
$$
\left(\frac{7}{3}\right)^{-12}
$$

But check the options:

(A) $-\left(\frac{3}{7}\right)^{-12}$ → wrong sign
(B) $\left(\frac{3}{7}\right)^{-12}$ → this is equal to $\left(\frac{7}{3}\right)^{12}$ → not same
(C) $-\left(\frac{7}{3}\right)^{-12}$ → wrong sign
(D) $\left(\frac{7}{3}\right)^{-12}$ → ✔ Correct!

So, yes.

✔ Answer: (D) $\left(\frac{7}{3}\right)^{-12}$

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Question 30:

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

We know:
$$
\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2
$$

Because:
$$
\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
$$

So:
$$
\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2
$$

Now check options:

(A) $\left(\frac{3}{4}\right)^2$ → wrong sign
(B) $-\left(\frac{3}{4}\right)^2$ → wrong
(C) $\left(\frac{4}{3}\right)^2$ → ✔ Correct
(D) $-\left(\frac{4}{3}\right)^2$ → wrong sign

✔ Answer: (C) $\left(\frac{4}{3}\right)^2$

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✔ Final Answers:

| Q.No | Answer |
|------|--------|
| 25 | (A) $4^{-3}$ |
| 26 | (B) $\left(\frac{1}{3}\right)^{-5}$ |
| 27 | (D) $\left(\frac{5}{3}\right)^{-4}$ |
| 28 | (C) $\left(\frac{3}{2}\right)^{-12}$ |
| 29 | (D) $\left(\frac{7}{3}\right)^{-12}$ |
| 30 | (C) $\left(\frac{4}{3}\right)^2$ |

Let me know if you'd like these explained in simpler terms or with diagrams!