• Factorize the given radicals and simplify each phrase.

### Example1

Simplify: $$5 \ \sqrt{2} \ + \ 7 \ \sqrt{32}$$

Solution:

$$5 \ \sqrt{2} \ + \ 7 \ \sqrt{32} \ = \ 5 \ \sqrt{2} \ + \ 7 \times 4 \ \sqrt{2} \ = \ 5 \ \sqrt{2} \ + \ 28 \ \sqrt{2}$$
$$5 \ \sqrt{2} \ + \ 28 \ \sqrt{2} \ = \ (5 \ + \ 28) \ \sqrt{2} \ = \ 33 \ \sqrt{2}$$

### Example2

Simplify: $$6 \ \sqrt{27} \ - \ 11 \ \sqrt{3}$$

Solution:

$$6 \ \sqrt{27} \ - \ 11 \ \sqrt{3} \ = \ 6 \times 3 \ \sqrt{3} \ - \ 11 \ \sqrt{3} \ = \ 18 \ \sqrt{3} \ - \ 11 \ \sqrt{3}$$
• Then, combine like terms:
$$18 \ \sqrt{3} \ - \ 11 \ \sqrt{3} \ = \ (18 \ - \ 11) \ \sqrt{3} \ = \ 7 \ \sqrt{3}$$

1) Find the answer: $$\sqrt{5 \ n^3} \ - \ \sqrt{5 \ n} \ =$$

2) Find the answer: $$\sqrt{14 \ q^3} \ + \ \sqrt{56 \ q^5} \ =$$

3) Find the answer: $$\sqrt{8 \ p^7} \ + \ \sqrt{72 \ p^9} \ =$$

4) Find the answer: $$\sqrt{180 \ x^3} \ - \ \sqrt{45 \ x} \ =$$

5) Find the answer: $$\sqrt{6 \ z^5} \ + \ \sqrt{96 \ z^3} \ =$$

6) Find the answer: $$\sqrt{75 \ t^{13}} \ - \ \sqrt{3 \ t^3} \ =$$

7) Find the answer: $$\sqrt{117 \ y^{23}} \ - \ \sqrt{52 \ t^{11}} \ =$$

8) Find the answer: $$\sqrt{175 \ n^{17}} \ + \ \sqrt{112 \ n^3} \ =$$

9) Find the answer: $$-2 \ \sqrt{60 \ q^9} \ + \ q^4 \ \sqrt{135 \ q} \ =$$

10) Find the answer: $$6 \ p^7 \sqrt{44 \ p^3} \ - \ 5 \ p^5 \sqrt{99 \ p^7} \ =$$

1) Find the answer: $$\sqrt{5 \ n^3} \ - \ \sqrt{5 \ n} \ =$$

$$\color{red}{\sqrt{5 \ n^3} \times \sqrt{5 \ n} \ = \ n \ \sqrt{5 \ n} \ - \ \sqrt{5 \ n} \ = \ (n \ - \ 1) \ \sqrt{5 \ n}}$$

2) Find the answer: $$\sqrt{14 \ q^3} \ + \ \sqrt{56 \ q^5} \ =$$

$$\color{red}{\sqrt{14 \ q^3} \ + \ \sqrt{56 \ q^5} \ = \ q \ \sqrt{14 \ q} \ + \ 2 \ q^2 \ \sqrt{14 \ q} \ = \ (q \ + \ 2 \ q^2) \ \sqrt{14 \ q}}$$

3) Find the answer: $$\sqrt{8 \ p^7} \ + \ \sqrt{72 \ p^9} \ =$$

$$\color{red}{\sqrt{8 \ p^7} \ + \ \sqrt{72 \ p^9} \ = \ 2 \ p^3 \ \sqrt{2 \ p} \ + \ 6 \ p^4 \ \sqrt{2 \ p} \ = \ (2 \ p^3 \ + \ 6 \ p^4) \ \sqrt{2 \ p}}$$

4) Find the answer: $$\sqrt{180 \ x^3} \ - \ \sqrt{45 \ x} \ =$$

$$\color{red}{\sqrt{180 \ x^3} \ - \ \sqrt{45 \ x} \ = \ 6 \ x \ \sqrt{5 \ x} \ - \ 3 \ \sqrt{5 \ x} \ = \ (6 \ x \ - \ 3) \ \sqrt{5 \ x}}$$

5) Find the answer: $$\sqrt{6 \ z^5} \ + \ \sqrt{96 \ z^3} \ =$$

$$\color{red}{\sqrt{6 \ z^5} \ + \ \sqrt{96 \ z^3} \ = \ z^2 \ \sqrt{6 \ z} \ + \ 4 \ z \ \sqrt{6 \ z} \ = \ (z^2 \ + \ 4 \ z) \ \sqrt{6 \ z}}$$

6) Find the answer: $$\sqrt{75 \ t^{13}} \ - \ \sqrt{3 \ t^3} \ =$$

$$\color{red}{\sqrt{75 \ t^{13}} \ - \ \sqrt{3 \ t^3} \ = \ 5 \ t^6 \ \sqrt{3 \ t} \ - \ t \ \sqrt{3 \ t} \ = \ (5 \ t^6 \ - \ t) \ \sqrt{3 \ t}}$$

7) Find the answer: $$\sqrt{117 \ y^{23}} \ - \ \sqrt{52 \ t^{11}} \ =$$

$$\color{red}{\sqrt{117 \ y^{23}} \ - \ \sqrt{52 \ y^{11}} \ = \ 3 \ y^{11} \ \sqrt{13 \ y} \ - \ 2 \ y^5 \ \sqrt{13 \ y} \ = \ (3 \ y^{11} \ - \ 2 \ y^5) \ \sqrt{13 \ y}}$$

8) Find the answer: $$\sqrt{175 \ n^{17}} \ + \ \sqrt{112 \ n^3} \ =$$

$$\color{red}{\sqrt{175 \ n^{17}} \ + \ \sqrt{112 \ n^3} \ = \ 5 \ n^8 \ \sqrt{7 \ n} \ + \ 4 \ n \ \sqrt{7 \ n} \ = \ (5 \ n^8 \ + \ 4 \ n) \ \sqrt{7 \ n}}$$

9) Find the answer: $$-2 \ \sqrt{60 \ q^9} \ + \ q^4 \ \sqrt{135 \ q} \ =$$

$$\color{red}{-2 \ \sqrt{60 \ q^9} \ + \ q^4 \ \sqrt{135 \ q} \ = \ 2 \ q^4 \ \sqrt{15 \ q} \ + \ 3 \ q^4 \ \sqrt{15 \ q} \ = \ (2 \ q^4 \ + \ 3 \ q^4) \ \sqrt{15 \ q}}$$ $$\color{red}{ \ = \ 5 \ q^4 \ \sqrt{15 \ q}}$$

10) Find the answer: $$6 \ p^7 \sqrt{44 \ p^3} \ - \ 5 \ p^5 \sqrt{99 \ p^7} \ =$$

$$\color{red}{6 \ p^7 \sqrt{44 \ p^3} \ - \ 5 \ p^5 \sqrt{99 \ p^7} \ = \ 12 \ p^8 \sqrt{11 \ p} \ - \ 15 \ p^8 \sqrt{11 \ p} \ = \ (12 \ p^8 \ - \ 15 \ p^8) \ \sqrt{11 \ p}}$$

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