Question

re - write and simplify the following expressions in radical form.
(a^{\frac{3}{4}})
(sqrt3{8a^4})
(sqrt{a^2})
(sqrt4{a^2})
(sqrt{a})

Explanation:

Step1: Recall the formula for converting exponents to radicals

The formula to convert a fractional exponent \(a^{\frac{m}{n}}\) to radical form is \( \sqrt[n]{a^{m}} \), where \(n\) is the index of the radical and \(m\) is the power of the base inside the radical. For the expression \(a^{\frac{1}{2}}\), we have \(m = 1\) and \(n=2\).

Step2: Apply the formula

Using the formula \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), substituting \(m = 1\) and \(n = 2\) into the formula for \(a^{\frac{1}{2}}\), we get \(a^{\frac{1}{2}}=\sqrt[2]{a^{1}}\), and \(\sqrt[2]{a}\) is usually written as \(\sqrt{a}\).

Answer:

\(\sqrt{a}\) (corresponding to the blue - colored option \(\sqrt{a^{2}}\) is incorrect, the correct radical form of \(a^{\frac{1}{2}}\) is \(\sqrt{a}\), assuming there is a typo and the intended expression for the blue option might have a different exponent, but based on the exponent \(a^{\frac{1}{2}}\), the radical form is \(\sqrt{a}\))