Question

simplify.
\sqrt{16y^{5}z^{11}}
assume that all variables represent positive real numbers.

Explanation:

Step1: Break down the radicand

We can express each part of the radicand as a product of perfect squares and remaining factors. For the coefficient, \(16 = 4^2\). For the variable \(y\), \(y^5=y^{4 + 1}=y^4\cdot y=(y^2)^2\cdot y\). For the variable \(z\), \(z^{11}=z^{10+1}=z^{10}\cdot z=(z^5)^2\cdot z\). So we have:
$$\sqrt{16y^{5}z^{11}}=\sqrt{4^{2}\cdot(y^{2})^{2}\cdot y\cdot(z^{5})^{2}\cdot z}$$

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\)

We can separate the perfect square factors from the non - perfect square factors:
$$\sqrt{4^{2}\cdot(y^{2})^{2}\cdot y\cdot(z^{5})^{2}\cdot z}=\sqrt{4^{2}}\cdot\sqrt{(y^{2})^{2}}\cdot\sqrt{(z^{5})^{2}}\cdot\sqrt{y\cdot z}$$

Step3: Simplify the square roots of perfect squares

We know that \(\sqrt{a^{2}}=a\) for \(a\geq0\). So \(\sqrt{4^{2}} = 4\), \(\sqrt{(y^{2})^{2}}=y^{2}\), \(\sqrt{(z^{5})^{2}}=z^{5}\). Then the expression becomes:
$$4\cdot y^{2}\cdot z^{5}\cdot\sqrt{yz}$$
$$ = 4y^{2}z^{5}\sqrt{yz}$$

Answer:

\(4y^{2}z^{5}\sqrt{yz}\)