Question

By what number of the following should \( \left(- \frac{3}{2} \right)^{-3} \) be divided so that the quotient may be equal to \( \left( \frac{4}{27} \right)^{-2} \)?

• A. -128
• B. 19683
• C. -132
• D. 16352

Hint:

When working with powers and quotients, simplify the expressions first and solve step-by-step to avoid confusion.

Answer 1

The Correct Option is B

We need to find a number \( x \) such that: \[ \left(- \frac{3}{2} \right)^{-3} \div x = \left( \frac{4}{27} \right)^{-2} \] Step 1: Simplify both sides. \[ \left(- \frac{3}{2} \right)^{-3} = \left( - \frac{2}{3} \right)^{3} = - \frac{8}{27} \] \[ \left( \frac{4}{27} \right)^{-2} = \left( \frac{27}{4} \right)^{2} = \frac{729}{16} \] Step 2: Now, solve for \( x \): \[ - \frac{8}{27} \div x = \frac{729}{16} \] \[ x = - \frac{8}{27} \times \frac{16}{729} \] \[ x = - \frac{128}{19683} \] Thus, the number that should divide is 19683.