Question

Use the laws of exponents to solve the following exponential equation.(x - 2)^1/2 = 9^1/4

Answer 1

To solve this question we will use the following properties of exponents:

$\begin{gathered} (a^{2m})^{\frac{1}{2m}}=|a|. \\ (a^{\frac{1}{m}})^m=a. \\ (a^m)^n=a^{mn.} \end{gathered}$

Now, notice that:

$\begin{gathered} 9=3^2, \\ \frac{1}{4}=\frac{1}{2}\cdot\frac{1}{2}\text{.} \end{gathered}$

Therefore:

$\begin{gathered} 9^{\frac{1}{4}}=9^{\frac{1}{2}\cdot\frac{1}{2}}=(9^{\frac{1}{2}})^{\frac{1}{2}} \\ =((3^2)^{\frac{1}{2}})^{\frac{1}{2}}=3^{\frac{1}{2}}. \end{gathered}$

Substituting the above result in the given equation we get:

$(x-2)^{\frac{1}{2}}=3^{\frac{1}{2}}.$

Taking the above equation to the power of 2 we get:

$\begin{gathered} ((x-2)^{\frac{1}{2}})^2=(3^{\frac{1}{2}})^2, \\ x-2=3. \end{gathered}$

Adding 2 to the above equation we get:

$\begin{gathered} x-2+2=3+2, \\ x=5. \end{gathered}$

Answer: Option b.