Question

revisiting exponents & their functions quick check
use the laws of exponents to solve for x in the equation \\(\frac{4^{\frac{1}{4}}}{x} = 4^{\frac{3}{4}}\\). (1 point)
\\(\circ\\) 2
\\(\circ\\) \\(\frac{1}{2}\\)
\\(\circ\\) 16
\\(\circ\\) \\(-\frac{1}{2}\\)

Explanation:

Step1: Simplify the left - hand side exponent

First, simplify the exponent in the numerator of the left - hand side. We know that when we have a fraction like \(\frac{a^{m}}{a^{n}}=a^{m - n}\), but here we have \(\frac{4^{\frac{1}{4}}}{x}=4^{\frac{3}{4}}\). Wait, actually, let's re - express the equation. The left - hand side is \(\frac{4^{\frac{1}{4}}}{x}\) and the right - hand side is \(4^{\frac{3}{4}}\). We can rewrite the equation as \(4^{\frac{1}{4}}\times\frac{1}{x}=4^{\frac{3}{4}}\), then \(\frac{1}{x}=\frac{4^{\frac{3}{4}}}{4^{\frac{1}{4}}}\).

Using the law of exponents \(a^{m}\div a^{n}=a^{m - n}\), for \(a = 4\), \(m=\frac{3}{4}\) and \(n=\frac{1}{4}\), we have \(\frac{4^{\frac{3}{4}}}{4^{\frac{1}{4}}}=4^{\frac{3}{4}-\frac{1}{4}}=4^{\frac{3 - 1}{4}}=4^{\frac{2}{4}}=4^{\frac{1}{2}}\).

Since \(4^{\frac{1}{2}}=\sqrt{4} = 2\), so \(\frac{1}{x}=2\).

Step2: Solve for \(x\)

If \(\frac{1}{x}=2\), then by cross - multiplying, we get \(2x = 1\), and \(x=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\) (corresponding to the option \(\frac{1}{2}\))