Express the following in exponential form :
a. `(16)/(81)`
b. `(-27)/(1000)`

Answer

To express the given fractions in exponential form, we will perform the prime factorization of the numerators and denominators.

### a. Expressing `(16)/(81)` in exponential form:

1. **Prime Factorization of 16**:
- 16 can be factored into prime numbers:
- \( 16 = 2 \times 2 \times 2 \times 2 = 2^4 \)

2. **Prime Factorization of 81**:
- 81 can be factored into prime numbers:
- \( 81 = 3 \times 3 \times 3 \times 3 = 3^4 \)

3. **Writing the fraction in exponential form**:
- Now we can write the fraction using the prime factorization:
- \( \frac{16}{81} = \frac{2^4}{3^4} \)

4. **Combining the powers**:
- We can express this as:
- \( \frac{2}{3}^4 \)

So, the exponential form of \( \frac{16}{81} \) is \( \left(\frac{2}{3}\right)^4 \).

### b. Expressing `(-27)/(1000)` in exponential form:

1. **Prime Factorization of -27**:
- -27 can be factored into prime numbers:
- \( -27 = - (3 \times 3 \times 3) = -3^3 \)

2. **Prime Factorization of 1000**:
- 1000 can be factored into prime numbers:
- \( 1000 = 10 \times 10 \times 10 = (2 \times 5)^3 = 2^3 \times 5^3 \)

3. **Writing the fraction in exponential form**:
- Now we can write the fraction using the prime factorization:
- \( \frac{-27}{1000} = \frac{-3^3}{2^3 \times 5^3} \)

4. **Separating the negative sign**:
- We can express this as:
- \( -\frac{3^3}{(2 \times 5)^3} = -\frac{3^3}{10^3} \)

So, the exponential form of \( \frac{-27}{1000} \) is \( -\frac{3^3}{10^3} \).

### Final Answers:
- a. \( \left(\frac{2}{3}\right)^4 \)
- b. \( -\frac{3^3}{10^3} \)

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