ANSWER
x = 5/4
EXPLANATION
[tex]2^x \cdot 2^{x-2} = \sqrt{2}[/tex]
Note that [tex]\sqrt{a} = a^{\frac{1}{2}}[/tex] so [tex]\sqrt{2} = 2^{\frac{1}{2} }[/tex]
Note that on the left-hand side, we can use exponent properties for multiplying two powers of the same base together: [tex]a^x \cdot a^y = a^{x+y} [/tex]
[tex]\begin{aligned}
2^x \cdot 2^{x-2} &= \sqrt{2} \\
2^{x + (x-2)} &= 2^{\frac{1}{2}} \\
2^{2x - 2} &= 2^{\frac{1}{2}}
\end{aligned}[/tex]
We can now equate the exponents because both sides of the equation are of the same base with no other terms.
[tex]\begin{aligned}
2^{2x - 2} &= 2^{\frac{1}{2}} \\
2x - 2 &= \tfrac{1}{2} \\
2x &= \tfrac{1}{2} + 2 \\
2x &= \tfrac{5}{2} \\
x &= \tfrac{5}{4}
\end{aligned}[/tex]
The answer is x = 5/4. We can confirm this by using this value in the original equation to get a true statement.
