Problems on Surds

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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