Question

Solve the equation (1/4)^(y+1) = 64. What are the steps to this equation?

Answer 1

To solve the equation (1/4)^(y+1) = 64, express both sides with a base of 2, equate the exponents, and then solve the resulting linear equation to find that y = -4.

Step-by-step explanation:

To solve the equation (1/4)^(y+1) = 64, we must first recognize that 64 is a power of 2, specifically 2^6. Since the base on the left side of the equation is a power of 2 (1/4 is 2^-2), it's helpful to express the right side of the equation with the same base to facilitate comparison. To do this, we will follow these steps:

1. Write 64 as 2^6 since 64 = 2 * 2 * 2 * 2 * 2 * 2.
2. Since 1/4 is 2^-2, the equation becomes (2^-2)^(y+1) = 2^6.
3. Apply the rule of exponents that (a^b)^c = a^(b*c) to get 2^(-2*(y+1)) = 2^6.
4. Since the bases are now the same (base 2), we can equate the exponents: -2 * (y + 1) = 6.
5. Expand the left side of the equation: -2y - 2 = 6.
6. Add 2 to both sides: -2y = 8.
7. Divide both sides by -2 to solve for y: y = -4.

Thus, the solution to the equation is y = -4.