Question

ExploreWhat value of a makes both of these equations true? I don’t get this.

Answer 1

we have the equations

`\begin{gathered} a^2=2^a \\ a^4=4^a \end{gathered}`

Let

`x=a^2`

Remember that

`a^4=(a^2)^2=x^2`

substitute the new variable x in the above equations

`\begin{gathered} a^2=2^a \\ x=2^a\text{ --}\longrightarrow\text{ equation 1} \end{gathered}`
`\begin{gathered} a^4=4^a \\ x^2=4^a\text{ --}\longrightarrow\text{ equation 2} \end{gathered}`

substitute equation 1 in equation 2

`(2a)^2=4^a`

equate the base

2a=4 ------> a=4/2=2

equate the exponents

2=a ----> a=2

therefore

The value of a=2