Math Problem Statement

(3/2)^-3

Solution

To solve (32)3\left(\frac{3}{2}\right)^{-3}, follow these steps:

Step 1: Apply the Negative Exponent Rule

The negative exponent rule states that an=1ana^{-n} = \frac{1}{a^n}. So, we apply this to the expression:

(32)3=1(32)3\left(\frac{3}{2}\right)^{-3} = \frac{1}{\left(\frac{3}{2}\right)^3}

Step 2: Calculate the Cube of 32\frac{3}{2}

Now, we need to raise 32\frac{3}{2} to the power of 3:

(32)3=3323=278\left(\frac{3}{2}\right)^3 = \frac{3^3}{2^3} = \frac{27}{8}

Step 3: Substitute Back

Now, substitute 278\frac{27}{8} into the expression:

(32)3=1278=827\left(\frac{3}{2}\right)^{-3} = \frac{1}{\frac{27}{8}} = \frac{8}{27}

Final Answer:

(32)3=827\left(\frac{3}{2}\right)^{-3} = \frac{8}{27}

Would you like further clarification or details on any step?

Related Questions:

  1. What happens when we apply positive exponents to fractions like (32)3\left(\frac{3}{2}\right)^3?
  2. How can the laws of exponents be used to simplify expressions with negative exponents?
  3. How do you simplify expressions like (53)2\left(\frac{5}{3}\right)^{-2}?
  4. How would you handle fractional exponents, for example (32)1/2\left(\frac{3}{2}\right)^{1/2}?
  5. Can negative exponents be simplified differently when applied to whole numbers?

Tip: Remember, negative exponents always mean "take the reciprocal" of the base before applying the exponent.

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Math Problem Analysis

Mathematical Concepts

Exponents
Negative Exponents
Fractions

Formulas

a^(-n) = 1 / (a^n)
(a/b)^n = a^n / b^n

Theorems

Negative Exponent Rule

Suitable Grade Level

Grades 7-9