((81)/(16))^(-3/4)times((64)/(27))^(-1/3)=

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: ((81)/(16))^(-3/4) times ((64)/(27))^(-1/3)

This problem involves simplifying an expression with fractional exponents and negative signs. Let's break it down step-by-step.

Understanding the Rules

• Negative Exponents: A term raised to a negative exponent is equal to its reciprocal raised to the positive version of the exponent. For example, x⁻² = 1/x².
• Fractional Exponents: A fractional exponent like (1/n) indicates taking the nth root of the base. For example, x¹/² is the square root of x.

Applying the Rules

Let's apply these rules to our expression:

1. Dealing with Negative Exponents:

   • ((81)/(16))^(-3/4) = ( (16)/(81) )^(3/4)
   • ((64)/(27))^(-1/3) = ( (27)/(64) )^(1/3)

2. Simplifying the Fractional Exponents:

   • ( (16)/(81) )^(3/4) = ( (2⁴)/(3⁴) )^(3/4) = (2/3)³
   • ( (27)/(64) )^(1/3) = ( (3³)/(4³) )^(1/3) = (3/4)

3. Multiplying the Simplified Terms:

   • (2/3)³ × (3/4) = (8/27) × (3/4) = 2/9

Final Answer

Therefore, the simplified form of the expression ((81)/(16))^(-3/4) times ((64)/(27))^(-1/3) is 2/9.