Question

simplify the expression.

(-2)^{\frac{4}{3}} \cdot (-2)^{\frac{5}{3}}

(-2)^{\frac{4}{3}} \cdot (-2)^{\frac{5}{3}} = \square

Explanation:

⚡ Using what you learned: Integer and Rational Exponents

Step 1: Apply the product rule of exponents

When multiplying expressions with the same base, add their exponents:

$$ a^m \cdot a^n = a^{m+n} $$

For this expression, the base is \( -2 \):

$$ (-2)^{\frac{4}{3}} \cdot (-2)^{\frac{5}{3}} = (-2)^{\frac{4}{3} + \frac{5}{3}} $$

Step 2: Add the exponents

Combine the fractional exponents:

$$ \frac{4}{3} + \frac{5}{3} = \frac{9}{3} = 3 $$

Step 3: Simplify the expression

Evaluate the base raised to the simplified exponent:

$$ (-2)^3 = (-2) \cdot (-2) \cdot (-2) = -8 $$

Answer:

$$ -8 $$