Question

Must answer in integers and reduced fractions only. No decimals

Answer 1

We must solve the following equation:

`8^(x+2)=4.`

1) First, we express the 8 and the 4 as a powers of 2:

`\begin{gathered} 8=2^3, \\ 4=2^2\text{.} \end{gathered}`

Replacing these equations in the equation above, we have:

`(2^3)^(x+2)=2^2.`

2) Using the property that the exponents multiplies we have:

`2^(3\cdot(x+2))=2^2.`

3) Because the basis are equal, the exponents must be equal too, so:

`3\cdot(x+2)=2.`

4) Finally, we solve the last equation for x:

`\begin{gathered} 3\cdot(x+2)=2, \\ 3x+6=2, \\ 3x=2-6, \\ 3x=-4, \\ x=-(4)/(3)\text{.} \end{gathered}`

Answer: x = -4/3

Summary:

`\begin{gathered} 8^(x+2)=4 \\ \Leftrightarrow(2^3)^(x+2)=2^2 \\ \Leftrightarrow2^(3\cdot(x+2))=2^2 \\ \Leftrightarrow3\cdot(x+2)=2 \\ \Leftrightarrow3x=-4 \\ \Leftrightarrow x=-(4)/(3) \end{gathered}`