Question

assuming x and y are both positive, write the following expression in simplest radical form. 4x²√(25x³y⁷)

Explanation:

Step1: Simplify the square root term

First, simplify \(\sqrt{25x^{3}y^{7}}\). We know that \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)).

Break down the radicand: \(25x^{3}y^{7}=25\cdot x^{2}\cdot x\cdot y^{6}\cdot y\).

Then, \(\sqrt{25x^{3}y^{7}}=\sqrt{25}\cdot\sqrt{x^{2}}\cdot\sqrt{x}\cdot\sqrt{y^{6}}\cdot\sqrt{y}\).

Simplify each square root: \(\sqrt{25} = 5\), \(\sqrt{x^{2}}=x\) (since \(x>0\)), \(\sqrt{y^{6}} = y^{3}\) (since \(y>0\)). So, \(\sqrt{25x^{3}y^{7}}=5\cdot x\cdot y^{3}\cdot\sqrt{xy}=5x y^{3}\sqrt{xy}\).

Step2: Multiply with the outside term

Now, multiply this with the outside term \(4x^{2}\). So, \(4x^{2}\cdot5x y^{3}\sqrt{xy}\).

When multiplying variables with exponents, we use the rule \(a^{m}\cdot a^{n}=a^{m + n}\). For the \(x\) terms: \(x^{2}\cdot x=x^{2 + 1}=x^{3}\). The constants: \(4\cdot5 = 20\). The \(y^{3}\) and \(\sqrt{xy}\) remain as is.

So, \(4x^{2}\cdot5x y^{3}\sqrt{xy}=20x^{3}y^{3}\sqrt{xy}\).

Answer:

\(20x^{3}y^{3}\sqrt{xy}\)