(64/b^27)^-2/3

Jasonbradley

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Jun 16, 2024

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Simplifying (64/b^27)^-2/3

This article will guide you through simplifying the expression (64/b^27)^-2/3.

Understanding the Rules

To simplify this expression, we will utilize the following rules of exponents:

• Fractional exponents: a^(m/n) = (a^m)^(1/n) = (a^(1/n))^m. This means taking a power to a fractional exponent is the same as taking the root and then the power, or vice versa.
• Negative exponents: a^(-n) = 1/a^n. This means a negative exponent indicates taking the reciprocal of the base raised to the positive exponent.
• Exponent of a fraction: (a/b)^n = a^n/b^n. This means the exponent applies to both the numerator and the denominator.

Step-by-Step Simplification

1. Apply the exponent to both the numerator and the denominator:

   (64/b^27)^(-2/3) = 64^(-2/3) / (b^27)^(-2/3)

2. Simplify the numerator:

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

3. Simplify the denominator:

   • (b^27)^(-2/3) = b^(27 * -2/3) = b^(-18) = 1/b^18

4. Combine the results:

   • 64^(-2/3) / (b^27)^(-2/3) = (1/16) / (1/b^18)

5. Dividing by a fraction is the same as multiplying by its reciprocal:

   • (1/16) / (1/b^18) = (1/16) * (b^18/1) = b^18/16

Conclusion

Therefore, the simplified form of (64/b^27)^-2/3 is b^18/16.