4^(х-1) + 4^х = 64/5

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вопрос

To solve this equation, first simplify the left side by combining the terms with the same base (4):

4^(x-1) + 4^x = 4^x 4^(-1) + 4^x = (1/4) 4^x + 4^x
= (1/4 + 1) 4^x
= (5/4) 4^x

Now, we can rewrite the equation as:

(5/4) * 4^x = 64/5

To solve for x, we can rewrite 64/5 as a multiple of 4:

64/5 = 64/5 * 4/4 = 256/20

So the equation becomes:

(5/4) * 4^x = 256/20

Now, divide both sides by (5/4) to solve for 4^x:

4^x = (256/20) / (5/4)
4^x = (256/20) * (4/5)
4^x = 1024/100
4^x = 64/25

Now, we can rewrite 64/25 as a power of 4:

64/25 = (4^3) / (5^2)

So the equation becomes:

4^x = (4^3) / (5^2)

Now we can equate the exponents:

x = 3 - 2
x = 1

Therefore, the solution to the equation 4^(x-1) + 4^x = 64/5 is x = 1.