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Simplify the expression: (-2/3)⁻³ * (-2/3)⁻²

Mathematics

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Answer

−24332-\frac{243}{32}

Step-by-step solution

5 steps

  1. Apply the product rule for exponents, which states that am×an=am+na^m \times a^n = a^{m+n}.

    In this problem, the base is (−23)\left(-\frac{2}{3}\right), and the exponents are −3-3 and −2-2.

    (−23)−3×(−23)−2=(−23)−3+(−2)\left(-\frac{2}{3}\right)^{-3} \times \left(-\frac{2}{3}\right)^{-2} = \left(-\frac{2}{3}\right)^{-3 + (-2)}

  2. Add the exponents.

    (−23)−3−2=(−23)−5\left(-\frac{2}{3}\right)^{-3 - 2} = \left(-\frac{2}{3}\right)^{-5}

  3. Apply the rule for negative exponents, which states that a−n=1ana^{-n} = \frac{1}{a^n} or (ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n.

    (−23)−5=(−32)5\left(-\frac{2}{3}\right)^{-5} = \left(-\frac{3}{2}\right)^5

  4. Calculate the fifth power of (−32)\left(-\frac{3}{2}\right).

    (−32)5=(−3)5(2)5\left(-\frac{3}{2}\right)^5 = \frac{(-3)^5}{(2)^5} (−3)5=(−3)×(−3)×(−3)×(−3)×(−3)=−243(-3)^5 = (-3) \times (-3) \times (-3) \times (-3) \times (-3) = -243 (2)5=2×2×2×2×2=32(2)^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32

  5. Substitute the calculated values back into the expression.

    −24332\frac{-243}{32}

    The final answer is −24332-\frac{243}{32}.

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