Question

mixed exercises
simplify.

1. $sqrt3{16y^4z^{12}}$

Explanation:

Step1: Factor the terms inside the cube root

We can factor \(16\) as \(8\times2\), \(y^4\) as \(y^3\times y\), and \(z^{12}\) as \((z^4)^3\). So the expression inside the cube root becomes:
\(\sqrt[3]{8\times2\times y^3\times y\times (z^4)^3}\)

Step2: Apply the cube root property \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\)

Using the property of cube roots that \(\sqrt[3]{abc}=\sqrt[3]{a}\times\sqrt[3]{b}\times\sqrt[3]{c}\), we can separate the factors:
\(\sqrt[3]{8}\times\sqrt[3]{y^3}\times\sqrt[3]{(z^4)^3}\times\sqrt[3]{2y}\)

Step3: Simplify each cube root

We know that \(\sqrt[3]{8} = 2\), \(\sqrt[3]{y^3}=y\), and \(\sqrt[3]{(z^4)^3}=z^4\). Substituting these values in, we get:
\(2\times y\times z^4\times\sqrt[3]{2y}\)

Step4: Combine the simplified terms

Multiplying the constant and the variables together, we have:
\(2yz^4\sqrt[3]{2y}\)

Answer:

\(2yz^4\sqrt[3]{2y}\)