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((3)/(4))^(-6)times((4)/(3))^(-8)=((3)/(4))^(n)

2 min read Jun 16, 2024
((3)/(4))^(-6)times((4)/(3))^(-8)=((3)/(4))^(n)

Simplifying Exponents: Finding the Value of n

This article explores how to simplify the given equation:

(3/4)^(-6) * (4/3)^(-8) = (3/4)^n

We'll utilize the properties of exponents to solve for the value of n.

Understanding the Properties of Exponents

  • Negative Exponents: A number raised to a negative exponent is equivalent to its reciprocal raised to the positive version of that exponent. For example: x^(-a) = 1/x^(a)
  • Product of Powers: When multiplying exponents with the same base, you add the powers together. For example: x^(a) * x^(b) = x^(a+b)

Applying the Properties to Simplify

  1. Apply the negative exponent rule:

    • (3/4)^(-6) = (4/3)^6
    • (4/3)^(-8) = (3/4)^8

  2. Substitute the simplified terms into the original equation:

    • (4/3)^6 * (3/4)^8 = (3/4)^n

  3. Apply the product of powers rule:

    • (3/4)^(6+8) = (3/4)^n

  4. Simplify:

    • (3/4)^(14) = (3/4)^n

  5. Solve for n:

    • Since the bases are the same, we can equate the exponents: n = 14

Conclusion

Therefore, the value of n in the equation (3/4)^(-6) * (4/3)^(-8) = (3/4)^n is 14.

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