Question

assuming x and y are both positive, write the following expression in simplest radical form. y√(24x⁵y³)

Explanation:

Step1: Factor the radicand

First, we factor the number and the variables inside the square root. For the coefficient 24, we can write it as \(24 = 4\times6\), where 4 is a perfect square. For the variable \(x^5\), we can write it as \(x^5=x^4\times x\), and for \(y^3\), we can write it as \(y^3 = y^2\times y\). So, \(\sqrt{24x^{5}y^{3}}=\sqrt{4\times6\times x^{4}\times x\times y^{2}\times y}\).

Step2: Simplify the square root

Using the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\) and \(b\) are non - negative real numbers), we can split the square root:
\(\sqrt{4\times6\times x^{4}\times x\times y^{2}\times y}=\sqrt{4}\times\sqrt{x^{4}}\times\sqrt{y^{2}}\times\sqrt{6xy}\)
We know that \(\sqrt{4} = 2\), \(\sqrt{x^{4}}=x^{2}\) (since \(x>0\)) and \(\sqrt{y^{2}} = y\) (since \(y > 0\)). So, \(\sqrt{4}\times\sqrt{x^{4}}\times\sqrt{y^{2}}\times\sqrt{6xy}=2x^{2}y\sqrt{6xy}\)

Step3: Multiply by the outside factor

Now we have the original expression \(y\sqrt{24x^{5}y^{3}}\), and we just found that \(\sqrt{24x^{5}y^{3}}=2x^{2}y\sqrt{6xy}\). So we multiply \(y\) with \(2x^{2}y\sqrt{6xy}\):
\(y\times(2x^{2}y\sqrt{6xy})=2x^{2}y^{2}\sqrt{6xy}\)

Answer:

\(2x^{2}y^{2}\sqrt{6xy}\)