Question

question
select the expression that is equivalent to \\(\sqrt4{256x^{16}y^{18}}\\) given \\(x > 0\\) and \\(y > 0\\).

answer attempt 1 out of 2
\\(\circ\\) \\(4x^{4}y^{\frac{9}{2}}\\) \\(\circ\\) \\(64x^{4}y^{\frac{9}{2}}\\)
\\(\circ\\) \\(4x^{\frac{1}{4}}y^{\frac{2}{9}}\\) \\(\circ\\) \\(64x^{\frac{1}{4}}y^{\frac{2}{9}}\\)

Explanation:

Step1: Simplify the fourth root of 256

We know that \(4^4 = 256\), so \(\sqrt[4]{256}=\sqrt[4]{4^4} = 4\) (since \(x>0\) and we are dealing with real - valued roots for positive numbers, the fourth root of a positive number is positive).

Step2: Simplify the fourth root of \(x^{16}\)

Using the property of exponents \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), for \(\sqrt[4]{x^{16}}\), we have \(n = 4\) and \(m = 16\). So \(\sqrt[4]{x^{16}}=x^{\frac{16}{4}}=x^{4}\) (because when taking the \(n\)th root of \(a^m\), we divide the exponent \(m\) by \(n\)).

Step3: Simplify the fourth root of \(y^{18}\)

Again, using the property \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), for \(\sqrt[4]{y^{18}}\), \(n = 4\) and \(m = 18\). So \(\sqrt[4]{y^{18}}=y^{\frac{18}{4}}=y^{\frac{9}{2}}\) (we simplify the fraction \(\frac{18}{4}\) by dividing both the numerator and the denominator by 2).

Step4: Combine the simplified terms

Now, we combine the results from the previous steps. \(\sqrt[4]{256x^{16}y^{18}}=\sqrt[4]{256}\times\sqrt[4]{x^{16}}\times\sqrt[4]{y^{18}}\). Substituting the values we found: \(4\times x^{4}\times y^{\frac{9}{2}}=4x^{4}y^{\frac{9}{2}}\)

Answer:

\(4x^{4}y^{\frac{9}{2}}\) (corresponding to the first option: \(4x^{4}y^{\frac{9}{2}}\))