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(2^3)^4=(2^2)^x

2 min read Jun 16, 2024
(2^3)^4=(2^2)^x

Solving for x in (2^3)^4 = (2^2)^x

This problem involves simplifying exponents and then solving for an unknown exponent. Let's break it down step by step:

Understanding the Properties of Exponents

  • Power of a Power: When raising a power to another power, we multiply the exponents. This means (a^m)^n = a^(m*n).
  • Equality of Exponents: If two exponential expressions have the same base and are equal, their exponents must also be equal.

Applying the Properties

  1. Simplify the left side: (2^3)^4 = 2^(3*4) = 2^12

  2. Simplify the right side: (2^2)^x = 2^(2*x)

  3. Set the expressions equal: 2^12 = 2^(2*x)

  4. Solve for x: Since the bases are the same, we can equate the exponents: 12 = 2*x

    Dividing both sides by 2, we get: x = 6

Conclusion

Therefore, the solution to the equation (2^3)^4 = (2^2)^x is x = 6.

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