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(64)^-2/3 X (1/4)^-2

2 min read Jun 16, 2024
(64)^-2/3 X (1/4)^-2

Simplifying the Expression (64)^-2/3 x (1/4)^-2

This article will guide you through the steps of simplifying the expression (64)^-2/3 x (1/4)^-2.

Understanding the Properties of Exponents

To simplify this expression, we need to understand a few key properties of exponents:

  • Negative Exponent: a^-n = 1/a^n
  • Fractional Exponent: a^(m/n) = (a^(1/n))^m = (n√a)^m
  • Product of Powers: a^m x a^n = a^(m+n)

Simplifying (64)^-2/3

  1. Apply the negative exponent property: (64)^-2/3 = 1/(64)^(2/3)
  2. Apply the fractional exponent property: 1/(64)^(2/3) = 1/(∛64)^2
  3. Calculate the cube root of 64: 1/(∛64)^2 = 1/(4)^2
  4. Simplify: 1/(4)^2 = 1/16

Simplifying (1/4)^-2

  1. Apply the negative exponent property: (1/4)^-2 = 1/((1/4)^2)
  2. Calculate the square of 1/4: 1/((1/4)^2) = 1/(1/16)
  3. Simplify by dividing by a fraction: 1/(1/16) = 16

Combining the Results

Now that we've simplified both parts of the expression, we can multiply them together:

(64)^-2/3 x (1/4)^-2 = (1/16) x 16 = 1

Therefore, the simplified form of the expression (64)^-2/3 x (1/4)^-2 is 1.

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