Question

4\^{\Large x-10}=\left(\dfrac{1}{64}\right)\^{\Large 5x+2}

Answer 1

The equation involves exponential functions with a common base of 2. By rewriting both sides with a base of 2, the exponents can be equated, resulting in a simple algebraic equation that can be solved for x.

Step-by-step explanation:

The equation 4x-10 = (1/64)5x+2 can be simplified using exponent rules. Recognizing that 64 is 2 to the 6th power (26), we can rewrite 1/64 as 2-6. Because the bases are both 2 (since 4 is 22), we can equate the exponents and solve for x.

To solve the equation, rewrite the bases as 22 raised to the power of x-10 on the left and 2-6 raised to the power of 5x+2 on the right, resulting in 22x-20 = 2-30x-12. The exponents can be set equal because the bases are now the same (both 2), yielding a simple algebraic equation 2x-20 = -30x-12, which can be solved for x. Finding the value of x involves combining like terms and isolating x on one side of the equation.

Answer 2

To solve the equation 4^(x-10) = (1/64)^(5x+2), we can start by expressing both sides with the same base.

Since 4 can be written as 2^2 and 1/64 can be written as 2^(-6), we can rewrite the equation as:

(2^2)^(x-10) = (2^(-6))^(5x+2)

Using the exponentiation rule, we can simplify the equation to:

2^(2(x-10)) = 2^((-6)(5x+2))

Now, we can equate the exponents:

2(x-10) = -6(5x+2)

Expanding and simplifying:

2x - 20 = -30x - 12

Combining like terms:

32x = -8

Dividing both sides by 32:

x = -8/32

Simplifying further:

x = -1/4

Therefore, the value of x that satisfies the equation is -1/4.

Step-by-step explanation: