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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