Question

#1-13
name:
unit 6: radical functions
date: per:
homework 1: $n^{th}$ roots; simplifying radicals
for each value, give the number of possible roots, then list the roots.

1. 196, square root(s)
2. 216, cube root(s)
3. 625, fourth root(s)

evaluate each expression

1. $\sqrt{225}$
2. $\sqrt3{512}$
3. $\sqrt3{-64}$
4. $-\sqrt4{1,296}$

simplify each radical expression.

1. $5\sqrt{150}$
2. $-8\sqrt{63}$
3. $\sqrt3{875}$
4. $2\sqrt3{-54}$
5. $-3\sqrt4{567}$
6. $6\sqrt4{405}$

Explanation:

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Questions 1-3: Roots Count & Listing

Step1: Q1: Count square roots

Square roots of positive numbers have 2 roots.

Step2: Q1: List 196 square roots

$14^2=196$, $(-14)^2=196$

Step3: Q2: Count cube roots

All real numbers have 1 real cube root.

Step4: Q2: List 216 cube root

$6^3=216$

Step5: Q3: Count fourth roots

Positive numbers have 2 real fourth roots.

Step6: Q3: List 625 fourth roots

$5^4=625$, $(-5)^4=625$
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Questions 4-7: Evaluate Radicals

Step7: Q4: Evaluate square root

$\sqrt{225} = \sqrt{15^2}$

Step8: Q5: Evaluate cube root

$\sqrt[3]{512} = \sqrt[3]{8^3}$

Step9: Q6: Evaluate negative cube root

$\sqrt[3]{-64} = \sqrt[3]{(-4)^3}$

Step10: Q7: Evaluate negative fourth root

$-\sqrt[4]{1296} = -\sqrt[4]{6^4}$
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Questions 8-13: Simplify Radicals

Step11: Q8: Factor 150 for simplification

$5\sqrt{150} = 5\sqrt{25 \times 6}$

Step12: Q8: Simplify the radical

$5 \times \sqrt{25} \times \sqrt{6} = 5 \times 5 \times \sqrt{6}$

Step13: Q9: Factor 63 for simplification

$-8\sqrt{63} = -8\sqrt{9 \times 7}$

Step14: Q9: Simplify the radical

$-8 \times \sqrt{9} \times \sqrt{7} = -8 \times 3 \times \sqrt{7}$

Step15: Q10: Factor 875 for simplification

$\sqrt[3]{875} = \sqrt[3]{125 \times 7}$

Step16: Q10: Simplify the radical

$\sqrt[3]{125} \times \sqrt[3]{7} = 5\sqrt[3]{7}$

Step17: Q11: Factor -54 for simplification

$2\sqrt[3]{-54} = 2\sqrt[3]{-27 \times 2}$

Step18: Q11: Simplify the radical

$2 \times \sqrt[3]{-27} \times \sqrt[3]{2} = 2 \times (-3) \times \sqrt[3]{2}$

Step19: Q12: Factor 567 for simplification

$-3\sqrt[4]{567} = -3\sqrt[4]{81 \times 7}$

Step20: Q12: Simplify the radical

$-3 \times \sqrt[4]{81} \times \sqrt[4]{7} = -3 \times 3 \times \sqrt[4]{7}$

Step21: Q13: Factor 405 for simplification

$6\sqrt[4]{405} = 6\sqrt[4]{81 \times 5}$

Step22: Q13: Simplify the radical

$6 \times \sqrt[4]{81} \times \sqrt[4]{5} = 6 \times 3 \times \sqrt[4]{5}$

Answer:

1. Number of roots: 2; Roots: $14, -14$
2. Number of roots: 1; Root: $6$
3. Number of roots: 2; Roots: $5, -5$
4. $15$
5. $8$
6. $-4$
7. $-6$
8. $25\sqrt{6}$
9. $-24\sqrt{7}$
10. $5\sqrt[3]{7}$
11. $-6\sqrt[3]{2}$
12. $-9\sqrt[4]{7}$
13. $18\sqrt[4]{5}$