(1/8)^3 X 64^4/4^3

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (1/8)³ x 64⁴/4³

This article will guide you through simplifying the expression (1/8)³ x 64⁴/4³. We will utilize the rules of exponents to break down the problem and arrive at a concise solution.

Understanding the Problem

The expression involves various exponents and fractions. To simplify it, we need to apply the following exponent rules:

• Product of Powers: xᵃ * xᵇ = xᵃ⁺ᵇ
• Power of a Power: (xᵃ)ᵇ = xᵃᵇ
• Quotient of Powers: xᵃ / xᵇ = xᵃ⁻ᵇ

Step-by-Step Solution

1. Simplify the bases:

   • 1/8 can be expressed as 2⁻³ (since 8 = 2³)
   • 64 can be expressed as 2⁶ (since 64 = 2⁶)

2. Apply the Power of a Power rule:

   • (1/8)³ = (2⁻³)³ = 2⁻⁹
   • 64⁴ = (2⁶)⁴ = 2²⁴
   • 4³ = (2²)³ = 2⁶

3. Substitute simplified terms back into the expression: (1/8)³ x 64⁴/4³ = 2⁻⁹ x 2²⁴ / 2⁶

4. Apply the Product of Powers rule: 2⁻⁹ x 2²⁴ = 2⁻⁹⁺²⁴ = 2¹⁵

5. Apply the Quotient of Powers rule: 2¹⁵ / 2⁶ = 2¹⁵⁻⁶ = 2⁹

Final Solution

The simplified expression (1/8)³ x 64⁴/4³ is equivalent to 2⁹.