Let's solve the equation [tex]\((\sqrt{2})^{3x-1} = (\sqrt{4})^{x-2}\)[/tex] step-by-step.
1. Simplify the Exponents:
- The left side of the equation is [tex]\((\sqrt{2})^{3x-1}\)[/tex].
- The right side of the equation is [tex]\((\sqrt{4})^{x-2}\)[/tex].
2. Express the Square Roots as Powers of 2:
- [tex]\(\sqrt{2}\)[/tex] can be written as [tex]\(2^{1/2}\)[/tex].
- [tex]\(\sqrt{4}\)[/tex] can be written as [tex]\(2^{2/2} = 2^1\)[/tex].
Therefore, the equation becomes:
[tex]\[
(2^{1/2})^{3x-1} = (2^1)^{x-2}
\][/tex]
3. Simplify the Exponential Expressions:
- Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can rewrite the equation as:
[tex]\[
2^{(1/2)(3x-1)} = 2^{1(x-2)}
\][/tex]
This simplifies to:
[tex]\[
2^{(3x-1)/2} = 2^{x-2}
\][/tex]
4. Set the Exponents Equal:
- Since the bases (2) are the same, we can set the exponents equal to each other:
[tex]\[
\frac{3x - 1}{2} = x - 2
\][/tex]
5. Solve for x:
- First, eliminate the fraction by multiplying both sides of the equation by 2:
[tex]\[
3x - 1 = 2(x - 2)
\][/tex]
- Distribute the 2 on the right-hand side:
[tex]\[
3x - 1 = 2x - 4
\][/tex]
- Subtract 2x from both sides to isolate the 'x' term on one side:
[tex]\[
3x - 2x - 1 = -4
\][/tex]
Simplifies to:
[tex]\[
x - 1 = -4
\][/tex]
- Add 1 to both sides:
[tex]\[
x = -3
\][/tex]
Therefore, the solution to the equation [tex]\((\sqrt{2})^{3x-1} = (\sqrt{4})^{x-2}\)[/tex] is [tex]\(x = -3\)[/tex].
In conclusion, [tex]\(x = -3\)[/tex] is the value that satisfies the given equation.
