By which number should (-1/4)⁻¹ be multiplied so that the product is the…

For any non-zero rational number a, the exponent −1 gives its multiplicative inverse: a⁻¹ = 1/a. The reciprocal of any number b is the value 1/b, so that b × (1/b) = 1.

1. (-1/4)⁻¹ is the multiplicative inverse of -1/4, which is -4.

2. (-7/4)⁻¹ is the multiplicative inverse of -7/4, which is -4/7.

3. The reciprocal of (-7/4)⁻¹ is the reciprocal of -4/7, which is -7/4 — this is the target product.

4. Let x be the required number, so (-4) × x = -7/4.

5. Solving for x: x = (-7/4) ÷ (-4) = 7/16.

Check: (-4) × 7/16 = -28/16 = -7/4, which matches the target product, confirming the value of x.

Hence, (-1/4)⁻¹ must be multiplied by 7/16.