Question: Simply each expression. Assume all variables are positive.
1. \sqrt[3]{3z^2}
2. \sqrt{27y^3}
3. \sqrt[3]{\frac{x^9}{27}}
4. \sqrt{\frac{50}{4}}
Write each expression in radical form and simplify.
5. 16^{3/4}
6. 7^{1/3}
7. -1000^{2/3}
8. 4^{5/2}
Write each expression by using rational exponents. Assume all variables are positive.
9. \sqrt[4]{11^2}
10. \sqrt[3]{y^2}
11. \sqrt[5]{7}
12. -\sqrt{19^7}
1. **Problem:** Simplify $$\sqrt[3]{3z^2}$$.
**Step 1:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{3z^2} = \sqrt[3]{3} \cdot \sqrt[3]{z^2}$$.
**Step 2:** Since variables are positive, $$\sqrt[3]{z^2} = z^{2/3}$$.
**Answer:** $$\sqrt[3]{3} z^{2/3}$$.
2. **Problem:** Simplify $$\sqrt{27y^3}$$.
**Step 1:** Write as $$\sqrt{27} \cdot \sqrt{y^3}$$.
**Step 2:** Simplify $$\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}$$.
**Step 3:** Simplify $$\sqrt{y^3} = y^{3/2} = y^{1} y^{1/2} = y \sqrt{y}$$.
**Answer:** $$3 y \sqrt{3y}$$.
3. **Problem:** Simplify $$\sqrt[3]{\frac{x^9}{27}}$$.
**Step 1:** Write as $$\frac{\sqrt[3]{x^9}}{\sqrt[3]{27}}$$.
**Step 2:** Simplify $$\sqrt[3]{x^9} = x^{9/3} = x^3$$.
**Step 3:** Simplify $$\sqrt[3]{27} = 3$$.
**Answer:** $$\frac{x^3}{3}$$.
4. **Problem:** Simplify $$\sqrt{\frac{50}{4}}$$.
**Step 1:** Simplify inside the root: $$\frac{50}{4} = \frac{25 \cdot 2}{4} = \frac{25 \cdot 2}{4}$$.
**Step 2:** Write as $$\sqrt{\frac{25 \cdot 2}{4}} = \frac{\sqrt{25 \cdot 2}}{\sqrt{4}} = \frac{\sqrt{25} \sqrt{2}}{2} = \frac{5 \sqrt{2}}{2}$$.
**Answer:** $$\frac{5 \sqrt{2}}{2}$$.
5. **Problem:** Simplify $$16^{3/4}$$.
**Step 1:** Write $$16 = 2^4$$.
**Step 2:** Substitute: $$16^{3/4} = (2^4)^{3/4} = 2^{4 \cdot \frac{3}{4}} = 2^3 = 8$$.
**Answer:** $$8$$.
6. **Problem:** Simplify $$7^{1/3}$$.
**Step 1:** This is the cube root of 7.
**Answer:** $$\sqrt[3]{7}$$.
7. **Problem:** Simplify $$-1000^{2/3}$$.
**Step 1:** Simplify $$1000^{2/3}$$ first.
**Step 2:** Write $$1000 = 10^3$$.
**Step 3:** Then $$1000^{2/3} = (10^3)^{2/3} = 10^{3 \cdot \frac{2}{3}} = 10^2 = 100$$.
**Step 4:** Apply the negative sign: $$-100$$.
**Answer:** $$-100$$.
8. **Problem:** Simplify $$4^{5/2}$$.
**Step 1:** Write $$4 = 2^2$$.
**Step 2:** Substitute: $$4^{5/2} = (2^2)^{5/2} = 2^{2 \cdot \frac{5}{2}} = 2^5 = 32$$.
**Answer:** $$32$$.
9. **Problem:** Write $$\sqrt[4]{11^2}$$ using rational exponents.
**Answer:** $$11^{2/4} = 11^{1/2} = \sqrt{11}$$.
10. **Problem:** Write $$\sqrt[3]{y^2}$$ using rational exponents.
**Answer:** $$y^{2/3}$$.
11. **Problem:** Write $$\sqrt[5]{7}$$ using rational exponents.
**Answer:** $$7^{1/5}$$.
12. **Problem:** Write $$-\sqrt{19^7}$$ using rational exponents.
**Step 1:** Write $$\sqrt{19^7} = (19^7)^{1/2} = 19^{7/2}$$.
**Answer:** $$-19^{7/2}$$.