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
= \(\sqrt{5\cdot 10}\)
= \(\sqrt{5\cdot 5\cdot 2}\)
= 5√2, which is an irrational number. Hence, it is a surd.
(ii) √8 × √6
= \(\sqrt{8\cdot 6}\)
= \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 3}\)
= 4√3, which is an irrational number. Hence, it is a surd.
(iii) √27 × √3
= \(\sqrt{27\cdot 3}\)
= \(\sqrt{3\cdot 3\cdot 3\cdot 3}\)
= 3 × 3
= 9, which is a rational number. Hence, it is not a surd.
(iv) √16 × √4
= \(\sqrt{16\cdot 4}\)
= \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2}\)
= 2 × 2 × 2
= 8, which is a rational number. Hence, it is not a surd.
(v) 5√8 × 2√6
= 5 × 2 \(\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 3}\)
= 10 × 2 × 2 × √3
= 40√3, which is an irrational number. Hence, it is a surd.
(vi) √125 × √5
= \(\sqrt{125\cdot 5}\)
= \(\sqrt{5\cdot 5\cdot 5\cdot 5}\)
= 5 × 5
= 25, which is a rational number. Hence, it is not a surd.
(vii) √100 × √2
= \(\sqrt{100\cdot 2}\)
= \(\sqrt{2\cdot 2\cdot 5\cdot 5\cdot 2}\)
= 2 × 5 × √2
= 10√2, which is an irrational number. Hence, it is a surd.
(viii) 6√2 × 9√3
= 6 × 9 \(\sqrt{2\cdot 3}\)
= 54 × √6
= 54√6, which is an irrational number. Hence, it is a surd.
(ix) √120 × √45
= \(\sqrt{120\cdot 45}\)
= \(\sqrt{2\cdot 2\cdot 2\cdot 3\cdot 5\cdot 3\cdot 3\cdot 5}\)
= 2 × 3 × 5 × √6
= 30√6, which is an irrational number. Hence, it is a surd.
(x) √15 × √6
= \(\sqrt{15\cdot 6}\)
= \(\sqrt{3\cdot 5\cdot 2\cdot 3}\)
= 3√10, which is an irrational number. Hence, it is a surd.
(xi) ∛5 × ∛25
= \(\sqrt[3]{5 × 25}\)
= \(\sqrt[3]{5 × 5 × 5}\)
= 5, which is a rational number. Hence, it is not a surd.
`2. Rationalize the denominator of the surd \(\frac{√5}{3√3}\).
Solution:
\(\frac{√5}{3√3}\)
= \(\frac{√5}{3√3}\) × \(\frac{√3}{√3}\)
= \(\frac{\sqrt{5 \times 3}}{3 \times \sqrt{3 \times 3}}\)
= \(\frac{√15}{3 × 3}\)
= \(\frac{1}{9}\)√15
3. Rationalize the denominator of the surd \(\frac{2}{√7  √3}\)
Solution:
\(\frac{2}{√7  √3}\)
= \(\frac{2 × (√7 + √3)}{(√7  √3) × (√7 + √3)}\)
= \(\frac{2 (√7 + √3)}{7  3}\)
= \(\frac{2 (√7 + √3)}{4}\)
= \(\frac{(√7 + √3)}{2}\)
4. Express the surd \(\frac{√3}{5√2}\) in the simplest form.
Solution:
\(\frac{√3}{5√2}\)
= \(\frac{√3}{5√2}\) × \(\frac{√2}{√2}\)
= \(\frac{\sqrt{3 \times 2}}{5 \times \sqrt{2 \times 2}}\)
= \(\frac{√6}{5 × 2}\)
= \(\frac{1}{10}\)√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) = x\(^{2}\)  y\(^{2}\)]
= 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) \(\sqrt[9]{19}\) × \(\sqrt[5]{10^{0}}\) = ____________.
(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) \(\sqrt[9]{19}\), [Since, we know, 10\(^{0}\) = 1]
(iv) mixed
(v) pure
(vi) 5√7
(vii) 4
(viii) 2
`11 and 12 Grade Math
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