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We will solve different types of problems on surds.
1. State whether the following are surds or not with reasons:
(i) √5 × √10
(ii) √8 × √6
(iii) √27 × √3
(iv) √16 × √4
(v) 5√8 × 2√6
(vi) √125 × √5
(vii) √100 × √2
(viii) 6√2 × 9√3
(ix)
√120 × √45
(x) √15 × √6
(xi) ∛5 × ∛25
Solution:
(i) √5 × √10
= √5⋅10
= √5⋅5⋅2
= 5√2, which is an irrational number. Hence, it is a surd.
(ii) √8 × √6
= √8⋅6
= √2⋅2⋅2⋅2⋅3
= 4√3, which is an irrational number. Hence, it is a surd.
(iii) √27 × √3
= √27⋅3
= √3⋅3⋅3⋅3
= 3 × 3
= 9, which is a rational number. Hence, it is not a surd.
(iv) √16 × √4
= √16⋅4
= √2⋅2⋅2⋅2⋅2⋅2
= 2 × 2 × 2
= 8, which is a rational number. Hence, it is not a surd.
(v) 5√8 × 2√6
= 5 × 2 √2⋅2⋅2⋅2⋅3
= 10 × 2 × 2 × √3
= 40√3, which is an irrational number. Hence, it is a surd.
(vi) √125 × √5
= √125⋅5
= √5⋅5⋅5⋅5
= 5 × 5
= 25, which is a rational number. Hence, it is not a surd.
(vii) √100 × √2
= √100⋅2
= √2⋅2⋅5⋅5⋅2
= 2 × 5 × √2
= 10√2, which is an irrational number. Hence, it is a surd.
(viii) 6√2 × 9√3
= 6 × 9 √2⋅3
= 54 × √6
= 54√6, which is an irrational number. Hence, it is a surd.
(ix) √120 × √45
= √120⋅45
= √2⋅2⋅2⋅3⋅5⋅3⋅3⋅5
= 2 × 3 × 5 × √6
= 30√6, which is an irrational number. Hence, it is a surd.
(x) √15 × √6
= √15⋅6
= √3⋅5⋅2⋅3
= 3√10, which is an irrational number. Hence, it is a surd.
(xi) ∛5 × ∛25
= 3√5×25
= 3√5×5×5
= 5, which is a rational number. Hence, it is not a surd.
2. Rationalize the denominator of the surd √53√3.
Solution:
√53√3
= √53√3 × √3√3
= √5×33×√3×3
= √153×3
= 19√15
3. Rationalize the denominator of the surd 2√7−√3
Solution:
2√7−√3
= 2×(√7+√3)(√7−√3)×(√7+√3)
= 2(√7+√3)7−3
= 2(√7+√3)4
= (√7+√3)2
4. Express the surd √35√2 in the simplest form.
Solution:
√35√2
= √35√2 × √2√2
= √3×25×√2×2
= √65×2
= 110√6, is the required simplest form of the given surd.
5. Expand (2√2 - √6)(2√2 + √6), expressing the result in the simplest form of surd:
Solution:
(2√2 - √6)(2√2 + √6)
= (2√2)2 - (√6)2, [Since, (x + y)(x - y) = x2 - y2]
= 8 - 6
= 2
6. Fill in the blanks:
(i) Surds having the same irrational factors are called ____________ surds.
(ii) √50 is a surd of order ____________.
(iii) 9√19 × 5√100 = ____________.
(iv) 6√5 is a ____________ surd.
(v) √18 is a ____________ surd.
(vi) 2√7 + 3√7 = ____________.
(vii) The order of the surd 3∜5 is a ____________.
(viii) ∛4 × ∛2 in the simplest form is = ____________.
Solution:
(i) similar.
(ii) 2
(iii) 9√19, [Since, we know, 100 = 1]
(iv) mixed
(v) pure
(vi) 5√7
(vii) 4
(viii) 2
11 and 12 Grade Math
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