Simplify the following square root expressions :
1) √64 + √196
2) √40 + √160
3) 2√425 - 3√68
4) √243 - 5√12 + √27
5) -√117 - √52
6) (√17)(√51)
7) (√35)(2√15)
8) (14√117) ÷ (7√52)
9) (7√5)2
10) (√3)3 + √27
1. Answer :
√64 + √196
Because 64 and 196 are perfect squares, we can find the square root of 64 and 194 as shown below.
√64 = √(8 ⋅ 8) √64 = 8 |
√196 = √(14 ⋅ 14) √196 = 14 |
√64 + √196 = 8 + 14
= 22
2. Answer :
√40 + √160
Decompose 40 and 160 into prime factors using synthetic division.
√40 = √(2 ⋅ 2 ⋅ 2 ⋅ 5) = 2√10
√160 = √(2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5) = 4√10
√40 + √160 :
= 2√10 + 4√10
= 6√10
3. Answer :
2√425 - 3√68
Decompose 425 and 68 into prime factors using synthetic division.
√425 = √(5 ⋅ 5 ⋅ 17) √425 = 5√17 |
√68 = √(2 ⋅ 2 ⋅ 17) √68 = 2√17 |
2√425 - 3√68 :
= 2(5√17) - 3(2√17)
= 10√17 - 6√17
= 4√17
4. Answer :
√243 - 5√12 + √27
Decompose 243, 12 and 27 into prime factors using synthetic division.
√243 = √(3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3) = 9√3
√12 = √(2 ⋅ 2 ⋅ 3) = 2√3
√27 = √(3 ⋅ 3 ⋅ 3) = 3√3
√243 - 5√12 + √27 :
= 9√3 - 5(2√3) + 3√3
= 9√3 - 10√3 + 3√3
= 2√3
5. Answer :
-√117 - √52
Decompose 117 and 52 into prime factors using synthetic division.
√117 = √(3 ⋅ 3 ⋅ 13) = 3√13
√52 = √(2 ⋅ 2 ⋅ 13) = 2√13
-√117 - √52 :
= -3√13 - 2√13
= -5√13
6. Answer :
(√17)(√51)
Decompose 17 and 51 into prime factors.
Because 17 is a prime number, it can't be decomposed anymore. So, √17 has to be kept as it is.
√51 = √(3 ⋅ 17) = √3 ⋅ √17
(√17)(√51) :
= (√17)(√3 ⋅ √17)
= (√17 ⋅ √17)√3
= 17√3
7. Answer :
(√35)(2√15)
Decompose 35 and 15 into prime factors.
√35 = √(5 ⋅ 7) = √5 ⋅ √7
√15 = √(5 ⋅ 3) = √5 ⋅ √3
(√35)(2√15) :
= (√5 ⋅ √7) ⋅ 2(√5 ⋅ √3)
= 2(√5 ⋅ √5)(√7 ⋅ √3)
= 2(5)(√(7 ⋅ 3)
= 10√21
8. Answer :
(14√117) ÷ (7√52)
Decompose 117 and 52 into prime factors using synthetic division.
√117 = √(3 ⋅ 3 ⋅ 13) √117 = 3√13 |
√52 = √(2 ⋅ 2 ⋅ 13) √52 = 2√13 |
(14√117) ÷ (7√52) :
= 14(3√13) ÷ 7(2√13)
= 42√13 ÷ 14√13
= 42√13/14√13
= 3
9. Answer :
(7√5)2 = 7√5 ⋅ 7√5
= (7 ⋅ 7)(√5 ⋅ √5)
= (49)(5)
= 245
10. Answer :
(√3)3 + √27 = (√3 ⋅ √3 ⋅ √3) + √(3 ⋅ 3 ⋅ 3)
= (3 ⋅ √3) + 3√3
= 3√3 + 3√3
= 6√3
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