What is the solution to equation 1 over the square root of 8 = 4^{(m + 2)}?

Solution:

Given, the equation is 1/√8 = 4^{(m + 2)}

When solving exponential equations, first we have to find a suitable common base.

Now, 8 = 2³ and 4 = 2²

The above equation becomes 1/√2³ = 2^{2(m + 2)}

Using the property of powers

\(\sqrt[n]{a}\) = a^1/n

Now, 1/2^3/2 = 2^{(2m + 4)}

Using the property, 1/a^x = a^-x

2^-3/2 = 2^{2m + 4}

Now, we have the equality of 2 powers with an equal base.

We can write it as the equality of exponents as

-3/2 = 2m + 4

Grouping of common terms,

2m = -3/2 - 4

2m = (-3 - 8)/2

2m = -11/2

m = -11/4

m = -2 ¾

Therefore, the solution to the equation is -2 ¾

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What is the solution to equation 1 over the square root of 8 = 4^{(m + 2)}?

Summary:

The solution to equation 1 over the square root of 8 = 4^{(m + 2)} is -2 ¾