Question

simplifying radical expressions
simplify.

1. $sqrt{35x^{2}} =$
2. $sqrt{90x^{2}} =$
3. $sqrt3{8a} =$
4. $sqrt{100x^{3}} =$
5. $sqrt{125a} =$
6. $sqrt3{88w^{3}} =$
7. $sqrt{80x} =$
8. $sqrt{216v} =$
9. $sqrt3{125x} =$
10. $sqrt{64x^{5}} =$
11. $sqrt4{4x^{2}} =$
12. $sqrt3{54a^{2}} =$
13. $sqrt{405} =$
14. $sqrt{512p^{3}} =$
15. $sqrt{216m^{4}} =$
16. $sqrt{264x^{3}y^{3}} =$
17. $sqrt{49x^{3}y^{3}} =$
18. $sqrt{16a^{4}b^{3}} =$
19. $sqrt{20x^{3}y^{3}} =$
20. $sqrt3{216yx^{3}} =$
21. $3sqrt{75x^{2}} =$
22. $5sqrt{80x^{2}} =$
23. $sqrt3{256x^{2}y^{3}} =$
24. $sqrt3{343x^{4}y^{2}} =$
25. $4sqrt{125a} =$
26. $sqrt3{625xy} =$
27. $2sqrt{8x^{2}y^{3}r} =$
28. $4sqrt{36x^{2}y^{3}z^{4}} =$
29. $2sqrt3{512x^{3}y^{4}} =$
30. $5sqrt{64a^{2}b^{3}c^{5}} =$
31. $2sqrt3{125x^{6}y^{12}} =$

Explanation:

Step1: Factor out perfect squares/cubes

For each radical, split the radicand into a product of a perfect square (for square roots) or perfect cube (for cube roots) and a remaining factor.

Step2: Simplify the radical

Use the property $\sqrt[n]{ab}=\sqrt[n]{a}\cdot\sqrt[n]{b}$ and $\sqrt[n]{a^n}=a$ (for non-negative $a$) to simplify.

---

1) $\sqrt{35x^2}$

Step1: Split into square term

$\sqrt{35 \cdot x^2}$

Step2: Simplify radical

$x\sqrt{35}$

2) $\sqrt{90x^2}$

Step1: Factor perfect square

$\sqrt{9 \cdot 10 \cdot x^2}$

Step2: Simplify radical

$3x\sqrt{10}$

3) $\sqrt[3]{8a}$

Step1: Factor perfect cube

$\sqrt[3]{8 \cdot a}$

Step2: Simplify radical

$2\sqrt[3]{a}$

4) $\sqrt{100x^3}$

Step1: Factor perfect square

$\sqrt{100 \cdot x^2 \cdot x}$

Step2: Simplify radical

$10x\sqrt{x}$

5) $\sqrt{125a}$

Step1: Factor perfect square

$\sqrt{25 \cdot 5a}$

Step2: Simplify radical

$5\sqrt{5a}$

6) $\sqrt[3]{88w^3}$

Step1: Factor perfect cube

$\sqrt[3]{8 \cdot 11 \cdot w^3}$

Step2: Simplify radical

$2w\sqrt[3]{11}$

7) $\sqrt{80x}$

Step1: Factor perfect square

$\sqrt{16 \cdot 5x}$

Step2: Simplify radical

$4\sqrt{5x}$

8) $\sqrt{216v}$

Step1: Factor perfect square

$\sqrt{36 \cdot 6v}$

Step2: Simplify radical

$6\sqrt{6v}$

9) $\sqrt[3]{125x}$

Step1: Factor perfect cube

$\sqrt[3]{125 \cdot x}$

Step2: Simplify radical

$5\sqrt[3]{x}$

10) $\sqrt{64x^5}$

Step1: Factor perfect square

$\sqrt{64 \cdot x^4 \cdot x}$

Step2: Simplify radical

$8x^2\sqrt{x}$

11) $\sqrt{4x^2}$

Step1: Identify perfect square

$\sqrt{(2x)^2}$

Step2: Simplify radical

$2|x|$ (or $2x$ for non-negative $x$)

12) $\sqrt[3]{54a^2}$

Step1: Factor perfect cube

$\sqrt[3]{27 \cdot 2a^2}$

Step2: Simplify radical

$3\sqrt[3]{2a^2}$

13) $\sqrt{405}$

Step1: Factor perfect square

$\sqrt{81 \cdot 5}$

Step2: Simplify radical

$9\sqrt{5}$

14) $\sqrt{512p^3}$

Step1: Factor perfect square

$\sqrt{256 \cdot 2 \cdot p^2 \cdot p}$

Step2: Simplify radical

$16p\sqrt{2p}$

15) $\sqrt{216m^4}$

Step1: Factor perfect square

$\sqrt{36 \cdot 6 \cdot (m^2)^2}$

Step2: Simplify radical

$6m^2\sqrt{6}$

16) $\sqrt{264x^3y^3}$

Step1: Factor perfect square

$\sqrt{4 \cdot 66 \cdot x^2 \cdot y^2 \cdot xy}$

Step2: Simplify radical

$2xy\sqrt{66xy}$

17) $\sqrt{49x^3y^3}$

Step1: Factor perfect square

$\sqrt{49 \cdot x^2 \cdot y^2 \cdot xy}$

Step2: Simplify radical

$7xy\sqrt{xy}$

18) $\sqrt{16a^4b^3}$

Step1: Factor perfect square

$\sqrt{16 \cdot (a^2)^2 \cdot b^2 \cdot b}$

Step2: Simplify radical

$4a^2b\sqrt{b}$

19) $\sqrt{20x^3y^3}$

Step1: Factor perfect square

$\sqrt{4 \cdot 5 \cdot x^2 \cdot y^2 \cdot xy}$

Step2: Simplify radical

$2xy\sqrt{5xy}$

20) $\sqrt[3]{216yx^3}$

Step1: Factor perfect cube

$\sqrt[3]{216 \cdot x^3 \cdot y}$

Step2: Simplify radical

$6x\sqrt[3]{y}$

21) $3\sqrt{75x^2}$

Step1: Factor perfect square

$3\sqrt{25 \cdot 3 \cdot x^2}$

Step2: Simplify radical

$3 \cdot 5x\sqrt{3} = 15x\sqrt{3}$

22) $5\sqrt{80x^2}$

Step1: Factor perfect square

$5\sqrt{16 \cdot 5 \cdot x^2}$

Step2: Simplify radical

$5 \cdot 4x\sqrt{5} = 20x\sqrt{5}$

23) $\sqrt[3]{256x^2y^3}$

Step1: Factor perfect cube

$\sqrt[3]{64 \cdot 4 \cdot x^2 \cdot y^3}$

Step2: Simplify radical

$4y\sqrt[3]{4x^2}$

24) $\sqrt[3]{343x^4y^2}$

Step1: Factor perfect cube

$\sqrt[3]{343 \cdot x^3 \cdot x y^2}$

Step2: Simplify radical

$7x\sqrt[3]{xy^2}$

25) $4\sqrt{125a}$

Step1: Factor perfect square

$4\sqrt{25 \cdot 5a}$

Step2: Simplify radical

$4 \cdot 5\sqrt{5a} = 20\sqrt{5a}$

26) $\sqrt[3]{625xy}$

Step1: Factor perfect cube

$\sqrt[3]{125 \cdot 5xy}$

Step2: Simplify radical

$5\sqrt[3]{5xy}$

27) $2\sqrt{8x^2y^3r}$

Step1: Factor perfect square

$2\sqrt{4 \cdot 2 \cdot x^2 \cdot y^2 \cdot yr}$

Step2: Simplify radical

$2 \cdot 2xy\sqrt{2yr} = 4xy\sqrt{2yr}$

28) $4\sqrt{36x^2y^3z^4}$

Step1: Factor perfect square

$4\sqrt{36 \cdot x^2 \cdot y^2 \cdot z^4 \cdot y}$

Step2: Simplify radical

$4 \cdot 6xyz^2\sqrt{y} = 24xyz^2\sqrt{y}$

29) $2\sqrt[3]{512x^3y^4}$

Step1: Factor perfect cube

$2\sqrt[3]{512 \cdot x^3 \cdot y^3 \cdot y}$

Step2: Simplify radical

$2 \cdot 8xy\sqrt[3]{y} = 16xy\sqrt[3]{y}$

30) $5\sqrt{64a^2b^3c^5}$

Step1: Factor perfect square

$5\sqrt{64 \cdot a^2 \cdot b^2 \cdot c^4 \cdot bc}$

Step2: Simplify radical

$5 \cdot 8abc^2\sqrt{bc} = 40abc^2\sqrt{bc}$

31) $2\sqrt[3]{125x^6y^{12}}$

Step1: Factor perfect cube

$2\sqrt[3]{125 \cdot (x^2)^3 \cdot (y^4)^3}$

Step2: Simplify radical

$2 \cdot 5x^2y^4 = 10x^2y^4$

Answer:

1. $x\sqrt{35}$
2. $3x\sqrt{10}$
3. $2\sqrt[3]{a}$
4. $10x\sqrt{x}$
5. $5\sqrt{5a}$
6. $2w\sqrt[3]{11}$
7. $4\sqrt{5x}$
8. $6\sqrt{6v}$
9. $5\sqrt[3]{x}$
10. $8x^2\sqrt{x}$
11. $2|x|$
12. $3\sqrt[3]{2a^2}$
13. $9\sqrt{5}$
14. $16p\sqrt{2p}$
15. $6m^2\sqrt{6}$
16. $2xy\sqrt{66xy}$
17. $7xy\sqrt{xy}$
18. $4a^2b\sqrt{b}$
19. $2xy\sqrt{5xy}$
20. $6x\sqrt[3]{y}$
21. $15x\sqrt{3}$
22. $20x\sqrt{5}$
23. $4y\sqrt[3]{4x^2}$
24. $7x\sqrt[3]{xy^2}$
25. $20\sqrt{5a}$
26. $5\sqrt[3]{5xy}$
27. $4xy\sqrt{2yr}$
28. $24xyz^2\sqrt{y}$
29. $16xy\sqrt[3]{y}$
30. $40abc^2\sqrt{bc}$
31. $10x^2y^4$