Let's solve the given equation for [tex]\( x \)[/tex]:
[tex]\[ 2\left(\frac{1}{2}\right)^{-x} = \left(8 x^4\right)^{\frac{1}{2}} \cdot x^{-2} \][/tex]
First, simplify the left-hand side of the equation:
[tex]\[ 2 \left(\frac{1}{2}\right)^{-x} \][/tex]
The term [tex]\(\left(\frac{1}{2}\right)^{-x}\)[/tex] is equivalent to [tex]\(2^x\)[/tex]:
[tex]\[ 2 \cdot 2^x = 2^{x+1} \][/tex]
Now let's simplify the right-hand side:
[tex]\[ \left(8 x^4\right)^{\frac{1}{2}} \cdot x^{-2} \][/tex]
Start by simplifying [tex]\(\left(8 x^4\right)^{\frac{1}{2}}\)[/tex]:
[tex]\[ \left(8 x^4\right)^{\frac{1}{2}} = \sqrt{8 x^4} \][/tex]
Recall that [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{8 x^4} = \sqrt{8} \cdot \sqrt{x^4} \][/tex]
Since [tex]\(\sqrt{8} = 2\sqrt{2}\)[/tex] and [tex]\(\sqrt{x^4} = x^2\)[/tex]:
[tex]\[ \sqrt{8 x^4} = 2\sqrt{2} \cdot x^2 \][/tex]
The original right-hand side becomes:
[tex]\[ 2\sqrt{2} \cdot x^2 \cdot x^{-2} \][/tex]
Simplify by combining the [tex]\(x^2\)[/tex] and [tex]\(x^{-2}\)[/tex] terms:
[tex]\[ 2\sqrt{2} \cdot x^{2-2} = 2\sqrt{2} \cdot x^0 \][/tex]
Since [tex]\( x^0 = 1 \)[/tex]:
[tex]\[ 2\sqrt{2} \][/tex]
Now the equation is simplified to:
[tex]\[ 2^{x+1} = 2\sqrt{2} \][/tex]
Recognize that [tex]\(\sqrt{2}\)[/tex] can be written as [tex]\(2^{1/2}\)[/tex]:
[tex]\[ 2^{x+1} = 2 \cdot 2^{1/2} \][/tex]
Combine the exponents:
[tex]\[ 2^{x+1} = 2^{1 + 1/2} \][/tex]
[tex]\[ 2^{x+1} = 2^{3/2} \][/tex]
Now, since the bases are equal, the exponents must be equal:
[tex]\[ x + 1 = \frac{3}{2} \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{3}{2} - 1 \][/tex]
[tex]\[ x = \frac{3}{2} - \frac{2}{2} \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
Thus, the solution to the equation is:
[tex]\[ x = \frac{1}{2} \][/tex]
