# 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