(1/4)^x=256

Jasonbradley

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Jun 16, 2024

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Solving the Equation (1/4)^x = 256

This article will guide you through solving the exponential equation (1/4)^x = 256.

Understanding the Basics

• Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, 4^3 means 4 multiplied by itself three times (4 * 4 * 4 = 64).
• Fractional Exponents: A fractional exponent represents a root. For example, 4^(1/2) is the square root of 4, which is 2.

Solving the Equation

1. Express 256 as a power of 4: 256 is the same as 4^4.
2. Rewrite the equation: Now the equation becomes (1/4)^x = 4^4.
3. Express 1/4 as a negative power of 4: (1/4) is equal to 4^(-1).
4. Substitute: The equation now becomes (4^(-1))^x = 4^4.
5. Simplify using exponent rules: When raising a power to another power, we multiply the exponents. This gives us 4^(-x) = 4^4.
6. Equate the exponents: For the equation to be true, the exponents must be equal. Therefore, -x = 4.
7. Solve for x: Dividing both sides by -1, we get x = -4.

Solution

Therefore, the solution to the equation (1/4)^x = 256 is x = -4.