Certainly! Let's solve the equation step-by-step: [tex]\((\sqrt{2})^{3x-1} = (\sqrt{4})^{x-2}\)[/tex].
1. Rewrite [tex]\(\sqrt{4}\)[/tex] as 2:
Given that [tex]\(\sqrt{4} = 2\)[/tex], we can rewrite the equation as:
[tex]$(\sqrt{2})^{3x-1} = 2^{x-2}$[/tex]
2. Express [tex]\(\sqrt{2}\)[/tex] as a power of 2:
We know that [tex]\(\sqrt{2}\)[/tex] can be written as [tex]\(2^{1/2}\)[/tex]. Therefore, the left side of the equation becomes:
[tex]$(2^{1/2})^{3x-1} = 2^{x-2}$[/tex]
3. Use the properties of exponents to combine the powers on the left side:
When raising a power to another power, we multiply the exponents:
[tex]$2^{(1/2)(3x-1)} = 2^{x-2}$[/tex]
4. Simplify the exponent on the left side:
Multiply [tex]\(1/2\)[/tex] by [tex]\(3x-1\)[/tex] to get:
[tex]$2^{(3x/2 - 1/2)} = 2^{x-2}$[/tex]
5. Since the bases are the same, set the exponents equal to each other:
Given that both sides of the equation have the same base (2), we can set the exponents equal to each other:
[tex]$\frac{3x}{2} - \frac{1}{2} = x - 2$[/tex]
6. Eliminate the fraction by multiplying the entire equation by 2:
[tex]$3x - 1 = 2x - 4$[/tex]
7. Solve for [tex]\(x\)[/tex]:
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]$3x - 2x - 1 = -4$[/tex]
[tex]$x - 1 = -4$[/tex]
Add 1 to both sides:
[tex]$x = -3$[/tex]
So, the solution to the equation [tex]\((\sqrt{2})^{3x-1} = (\sqrt{4})^{x-2}\)[/tex] is:
[tex]$x = -3$[/tex]
