Power and Exponent Worksheet-1 featuring math problems on converting expressions to rational numbers with negative and positive exponents.

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 as required.

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

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

We want to write this using a negative exponent. Recall:
$$
\frac{1}{a^n} = a^{-n}
$$

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

✔ Answer: (A) $4^{-3}$

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

We are given:
$$
3^5 = 243
$$

We want to express this as a rational number with a negative exponent.

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

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

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

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

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

We want to express this using a negative exponent.

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

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

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

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

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

Apply power of a power:
$$
= \left(\frac{3}{2}\right)^{4 \times (-3)} = \left(\frac{3}{2}\right)^{-12}
$$

Now, we can rewrite this as:
$$
\left(\frac{3}{2}\right)^{-12} = \left(\frac{2}{3}\right)^{12}
$$

But the question asks for expression with a negative exponent — so we keep it as $\left(\frac{3}{2}\right)^{-12}$, which is already in negative exponent form.

But let’s check options:

(A) $-\left(\frac{3}{2}\right)^{-12}$ → wrong (has a minus sign)

(B) $\left(\frac{2}{3}\right)^{-12}$ → this is equal to $\left(\frac{3}{2}\right)^{12}$ → not same

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

(D) $-\left(\frac{2}{3}\right)^{-12}$ → wrong (negative sign)

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

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

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

This is already in negative exponent form.

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

But again, the question wants expression with a negative exponent, so $\left(\frac{7}{3}\right)^{-12}$ is fine.

But look at options:

(A) $-\left(\frac{3}{7}\right)^{-12}$ → has negative sign → wrong

(B) $\left(\frac{3}{7}\right)^{-12}$ → this equals $\left(\frac{7}{3}\right)^{12}$ → not equal

(C) $-\left(\frac{7}{3}\right)^{-12}$ → negative sign → wrong

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

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

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

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

We want to write it with a positive exponent.

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

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

✔ 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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🔍 Key Rules Used:

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

Let me know if you'd like these explained visually!