(4/b)^(-3) ((2a^2)/3b)^4

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (4/b)^(-3) ((2a^2)/3b)^4

This article will guide you through simplifying the expression (4/b)^(-3) ((2a^2)/3b)^4. We will break down the steps to make it easier to understand.

Understanding the Rules

Before we start simplifying, let's review some important exponent rules:

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

Applying the Rules

1. Simplify the Negative Exponent:

   • (4/b)^(-3) = 1/(4/b)^3

2. Apply the Fractional Exponent Rule:

   • 1/(4/b)^3 = 1/(4^3/b^3)
   • 1/(4^3/b^3) = b^3/4^3

3. Simplify the Second Term:

   • ((2a^2)/3b)^4 = (2^4(a^2)^4)/(3^4b^4)
   • (2^4(a^2)^4)/(3^4b^4) = 16a^8/81b^4

4. Combine the Simplified Terms:

   • (b^3/4^3) * (16a^8/81b^4) = (16a^8 * b^3) / (4^3 * 81 * b^4)

5. Simplify further:

   • (16a^8 * b^3) / (4^3 * 81 * b^4) = (16a^8) / (64 * 81 * b)
   • Final Simplified Expression: a^8 / (324b)

Conclusion

By applying the rules of exponents, we have successfully simplified the expression (4/b)^(-3) ((2a^2)/3b)^4 to a^8 / (324b). This simplified form is easier to work with and understand. Remember, it's essential to understand the exponent rules to confidently manipulate expressions like this.