Question: Simplify each expression. Assume all variables are positive.
1. ⁴√(32z⁷)
2. ³√(27y³)
3. ³√(x⁹ / 27)
4. √(50 / z)
1. Simplify $$\sqrt[4]{32z^{7}}$$
- Start by expressing 32 as a power of 2: $$32 = 2^{5}$$.
- Rewrite the expression: $$\sqrt[4]{2^{5} z^{7}}$$.
- Use the property $$\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$$:
$$2^{\frac{5}{4}} z^{\frac{7}{4}}$$.
- Separate integer and fractional exponents:
$$2^{1 + \frac{1}{4}} z^{1 + \frac{3}{4}} = 2 \cdot 2^{\frac{1}{4}} \cdot z \cdot z^{\frac{3}{4}}$$.
- Rewrite as:
$$2z \sqrt[4]{2} \sqrt[4]{z^{3}} = 2z \sqrt[4]{2z^{3}}$$.
2. Simplify $$\sqrt[3]{27y^{3}}$$
- Recognize $$27 = 3^{3}$$.
- Rewrite:
$$\sqrt[3]{3^{3} y^{3}} = \sqrt[3]{(3y)^{3}}$$.
- Cube root of a cube is the base:
$$3y$$.
3. Simplify $$\sqrt[3]{\frac{x^{9}}{27}}$$
- Rewrite denominator: $$27 = 3^{3}$$.
- Expression becomes:
$$\sqrt[3]{\frac{x^{9}}{3^{3}}} = \sqrt[3]{\frac{x^{9}}{3^{3}}}$$.
- Apply cube root to numerator and denominator separately:
$$\frac{\sqrt[3]{x^{9}}}{\sqrt[3]{3^{3}}} = \frac{x^{\frac{9}{3}}}{3^{\frac{3}{3}}} = \frac{x^{3}}{3}$$.
- Show cancellation step:
$$\frac{\cancel{\sqrt[3]{x^{9}}}}{\cancel{\sqrt[3]{3^{3}}}} = \frac{x^{3}}{3}$$.
4. Simplify $$\sqrt{\frac{50}{z}}$$
- Express 50 as $$25 \times 2$$:
$$\sqrt{\frac{25 \times 2}{z}} = \sqrt{\frac{25}{z} \times 2}$$.
- Use property $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$:
$$\frac{\sqrt{25 \times 2}}{\sqrt{z}} = \frac{\sqrt{25} \sqrt{2}}{\sqrt{z}} = \frac{5 \sqrt{2}}{\sqrt{z}}$$.
- Rationalize denominator:
$$\frac{5 \sqrt{2}}{\sqrt{z}} \times \frac{\sqrt{z}}{\sqrt{z}} = \frac{5 \sqrt{2z}}{z}$$.
Final answers:
1. $$2z \sqrt[4]{2z^{3}}$$
2. $$3y$$
3. $$\frac{x^{3}}{3}$$
4. $$\frac{5 \sqrt{2z}}{z}$$