Simplify the following radical expressions :
Question 1 :
√3 + √12
Question 2 :
√75 + √3
Question 3 :
√18 + √98
Question 4 :
√5 + √20 - √125
Question 5 :
√5 + 3√7 - 4√5 - 5√7
Question 6 :
3√3 + 4√3 - √2
Question 7 :
2(√5 - √3) + 3(√3 - √5)
Question 8 :
3√16 + 3√54
Question 9 :
√25 + 53√64
Question 10 :
83√686 - 53√250
Question 11 :
√(12w) + √(27w)
Question 12 :
√45y3 + √25y3
Question 13 :
3√8x3y6 + √9x2y4
Question 14 :
√4p2q4 - 3√125p3q6
1. Answer :
= √3 + √12
= √3 + √(2 ⋅ 2 ⋅ 3)
= √3 + 2√3
= 3√3
2. Answer :
= √75 + √3
= √(5 ⋅ 5 ⋅ 3) + √3
= 5√3 + √3
= 6√3
3. Answer :
= √18 + √98
= √(3 ⋅ 3 ⋅ 2) + √(7 ⋅ 7 ⋅ 2)
= 3√2 + 7√2
= 10√2
4. Answer :
= √5 + √20 - √125
= √5 + √(2 ⋅ 2 ⋅ 5) - √(5 ⋅ 5 ⋅ 5)
= √5 + 2√5 - 5√5
= -2√5
5. Answer :
√5 + 3√7 - 4√5 - 5√7
Group the like radicals.
= (√5 - 4√5) + (3√7 - 5√7)
Combine the like radicals.
= (-3√5) + (-2√7)
= -3√5 - 2√7
6. Answer :
= 3√3 + 4√3 - √2
Group the like radicals.
= (3√3 + 4√3) - √2
Combine the like radicals.
= 7√3 - √2
7. Answer :
= 2(√5 - √3) + 3(√3 - √5)
Use Distributive Property.
= 2√5 - 2√3 + 3√3 - 3√5
Group the like radicals.
= (2√5 - 3√5) + (-2√3 + 3√3)
Combine the like radicals.
= -√5 + √3
8. Answer :
= 3√16 + 3√54
= 3√(2 ⋅ 2 ⋅ 2 ⋅ 2) + 3√(3 ⋅ 3 ⋅ 3 ⋅ 2)
= 23√2 + 33√2
= 53√2
9. Answer :
= √25 + 53√64
= √(5 ⋅ 5) + 53√(4 ⋅ 4 ⋅ 4)
= 5 + 5(4)
= 5 + 20
= 25
10. Answer :
= 83√686 - 53√250
= 83√(7 ⋅ 7 ⋅ 7 ⋅ 2) - 53√(5 ⋅ 5 ⋅ 5 ⋅ 2)
= 8(73√2) - 5(53√2)
= 563√2 - 253√2
= 313√2
11. Answer :
= √(12w) + √(27w)
= √(2 ⋅ 2 ⋅ 3 ⋅ w) + √(3 ⋅ 3 ⋅ 3 ⋅ w)
= 2√3w + 3√3w
= 5√3w
12. Answer :
= √45y3 + √25y3
= √(3 ⋅ 3 ⋅ 5 ⋅ y ⋅ y ⋅ y) + √(2 ⋅ 2 ⋅ 5 ⋅ y ⋅ y ⋅ y)
= 3y√5y + 2y√5y
=5y√5y
= 5y√5y
13. Answer :
= 3√8x3y6 + √9x2y4
= 3√(2 ⋅ 2 ⋅ 2 ⋅ x ⋅ x ⋅ x ⋅ y2 ⋅ y2 ⋅ y2) + √(3 ⋅ 3 ⋅ x ⋅ x ⋅ y2 ⋅ y2)
= 2xy2 + 3xy2
= 5xy2
14. Answer :
= √4p2q4 - 3√125p3q6
= √(2 ⋅ 2 ⋅ p ⋅ p ⋅ q2 ⋅ q2) - √(5 ⋅ 5 ⋅ 5 ⋅ p ⋅ p ⋅ p ⋅ q2 ⋅ q2 ⋅ q2)
= 2pq2 - 5pq2
= - 3pq2
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