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Let's solve each question step by step from the worksheet on Power and Exponent, focusing on converting expressions to forms with negative exponents or positive exponents using exponent rules.

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

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

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

Recall:
$$
\frac{1}{a^n} = a^{-n}
$$

So,
$$
\frac{1}{64} = \frac{1}{4^3} = 4^{-3}
$$

✔ So the correct answer is:
(A) $4^{-3}$

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(26) Expression of $3^5$ as a rational number with a negative exponent is ________.

We have $3^5$, which is positive. We need to write it using a negative exponent.

We use:
$$
a^n = \frac{1}{a^{-n}} \quad \Rightarrow \quad 3^5 = \frac{1}{3^{-5}}
$$

But that’s not in the form of a rational number with a negative exponent directly.

Wait — let’s think: can we write $3^5$ as something like $(\text{fraction})^{-n}$?

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

✔ So the expression with a negative exponent is:
(B) $\left(\frac{1}{3}\right)^{-5}$

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

We have:
$$
\left(\frac{3}{5}\right)^4 = \frac{3^4}{5^4} = \frac{81}{625}
$$

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

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

✔ So the correct answer is:
(D) $\left(\frac{5}{3}\right)^{-4}$

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

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

Using the rule $(a^m)^n = a^{m \cdot n}$

Now, $\left(\frac{3}{2}\right)^{-12} = \left(\frac{2}{3}\right)^{12}$

But the question asks for expression with a negative exponent.

So we keep it as $\left(\frac{3}{2}\right)^{-12}$, but look at options:

- (A) $-\left(\frac{3}{2}\right)^{-12}$ → has a negative sign, not correct
- (B) $\left(\frac{2}{3}\right)^{-12}$ → This is $\left(\frac{3}{2}\right)^{12}$, not same
- (C) $\left(\frac{3}{2}\right)^{-12}$ → ✔ matches exactly
- (D) $-\left(\frac{2}{3}\right)^{-12}$ → incorrect

✔ So the correct answer is:
(C) $\left(\frac{3}{2}\right)^{-12}$

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

But again, we are to express it with a negative exponent.

So we leave it as $\left(\frac{7}{3}\right)^{-12}$, or equivalently:

But check options:
- (A) $-\left(\frac{3}{7}\right)^{-12}$ → has negative sign
- (B) $\left(\frac{3}{7}\right)^{-12}$ → This is $\left(\frac{7}{3}\right)^{12}$ → not same
- (C) $-\left(\frac{7}{3}\right)^{-12}$ → has negative sign
- (D) $\left(\frac{7}{3}\right)^{-12}$ → ✔ Correct!

So the expression is already $\left(\frac{7}{3}\right)^{-12}$

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

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

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

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) $-\left(\frac{3}{4}\right)^2$ → negative
- (C) $\left(\frac{4}{3}\right)^2$ → ✔ Correct
- (D) $-\left(\frac{4}{3}\right)^2$ → negative

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

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✔ Final Answers:

| Question | 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) $\left(\frac{4}{3}\right)^2$ |

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🔍 Summary of Key Rules Used:

1. $a^{-n} = \frac{1}{a^n}$
2. $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$
3. $(a^m)^n = a^{m \cdot n}$
4. Negative exponent means reciprocal; positive exponent means normal fraction.

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