Ex. 4.3 – Class 7 and 8 Resources Center

Exercise – 4.3

Q1. Express each of the following as mixed radicals.

(a) Express \( \sqrt{18} \) as a mixed radical.

\[ \sqrt{18} =\sqrt{9\times2} =\sqrt9\times\sqrt2 =3\sqrt2 \]

Answer: \(3\sqrt2\)

(b) Express \( \sqrt{45} \) as a mixed radical.

\[ \sqrt{45} =\sqrt{9\times5} =\sqrt9\times\sqrt5 =3\sqrt5 \]

Answer: \(3\sqrt5\)

(c) Express \( \sqrt{405} \) as a mixed radical.

\[ \sqrt{405} =\sqrt{81\times5} =\sqrt{81}\times\sqrt5 =9\sqrt5 \]

Answer: \(9\sqrt5\)

(d) Express \( \sqrt{300} \) as a mixed radical.

\[ \sqrt{300} =\sqrt{100\times3} =\sqrt{100}\times\sqrt3 =10\sqrt3 \]

Answer: \(10\sqrt3\)

(e) Express \( \sqrt{180} \) as a mixed radical.

\[ \sqrt{180} =\sqrt{36\times5} =\sqrt{36}\times\sqrt5 =6\sqrt5 \]

Answer: \(6\sqrt5\)

Q2. Express each of the following as pure radicals.

(a) Express \(2\sqrt6\) as a pure radical.

\[ 2\sqrt6 =\sqrt4\times\sqrt6 =\sqrt{24} \]

Answer: \(\sqrt{24}\)

(b) Express \(5\sqrt7\) as a pure radical.

\[ 5\sqrt7 =\sqrt{25}\times\sqrt7 =\sqrt{175} \]

Answer: \(\sqrt{175}\)

(c) Express \(4\sqrt5\) as a pure radical.

\[ 4\sqrt5 =\sqrt{16}\times\sqrt5 =\sqrt{80} \]

Answer: \(\sqrt{80}\)

(d) Express \(6\sqrt{10}\) as a pure radical.

\[ 6\sqrt{10} =\sqrt{36}\times\sqrt{10} =\sqrt{360} \]

Answer: \(\sqrt{360}\)

(e) Express \(10\sqrt{13}\) as a pure radical.

\[ 10\sqrt{13} =\sqrt{100}\times\sqrt{13} =\sqrt{1300} \]

Answer: \(\sqrt{1300}\)

Exercise 4.3 – Part 2

Q3. Simplify.

(a) Simplify \( \sqrt6 \times \sqrt2 \).

\[ \sqrt6\times\sqrt2 =\sqrt{6\times2} =\sqrt{12} =\sqrt{4\times3} =2\sqrt3 \]

Answer: \(2\sqrt3\)

(b) Simplify \( \sqrt5 \times \sqrt{30} \times \sqrt8 \times \sqrt3 \).

\[ \sqrt5\times\sqrt{30}\times\sqrt8\times\sqrt3 =\sqrt{5\times30\times8\times3} =\sqrt{3600} =60 \]

Answer: \(60\)

(c) Simplify \(8\sqrt3 \times 7\sqrt5 \div 14\sqrt{15}\).

\[ =\frac{8\sqrt3\times7\sqrt5}{14\sqrt{15}} \] \[ =\frac{56\sqrt{15}}{14\sqrt{15}} =\frac{56}{14} =4 \]

Answer: \(4\)

(d) Simplify \( \sqrt{96} \div \sqrt{12} \).

\[ \frac{\sqrt{96}}{\sqrt{12}} =\sqrt{\frac{96}{12}} =\sqrt8 =\sqrt{4\times2} =2\sqrt2 \]

Answer: \(2\sqrt2\)

(e) Simplify \( (3\sqrt2+5)(3\sqrt2-5) \).

\[ (a+b)(a-b)=a^2-b^2 \] \[ =(3\sqrt2)^2-5^2 \] \[ =9\times2-25 =18-25 =-7 \]

Answer: \(-7\)

Exercise 4.3 – Part 3

Q4. Simplify.

(a) Simplify

\[ \frac{\sqrt{72}\times\sqrt{192}\times\sqrt{242}} {\sqrt{363}\times\sqrt{48}\times\sqrt{108}} \] \[ =\sqrt{\frac{72\times192\times242}{363\times48\times108}} \] \[ =\sqrt{\frac{(3^2\times2^3)(3\times2^6)(11^2\times2)} {(11\times3\times11)(3\times2^4)(2^2\times3^3)}} \] \[ =\sqrt{\frac{2^6}{3^2}} =\sqrt{\frac{64}{9}} =\frac{8}{3} \]

Answer: \(\dfrac{8}{3}\)

(b) Simplify \( \sqrt8\times\sqrt{30}\times\sqrt3\times\sqrt5 \).

\[ =\sqrt{8\times30\times3\times5} =\sqrt{3600} =60 \]

Answer: \(60\)

(c) Simplify \( \sqrt{40}+\sqrt{90}-\sqrt{18}-\sqrt{160}+\sqrt{32} \).

\[ =2\sqrt{10}+3\sqrt{10}-3\sqrt2-4\sqrt{10}+4\sqrt2 \] \[ =(2+3-4)\sqrt{10}+(-3+4)\sqrt2 \] \[ =\sqrt{10}+\sqrt2 \]

Answer: \(\sqrt{10}+\sqrt2\)

(d) Simplify \( (\sqrt5+\sqrt2)^2+(\sqrt5-\sqrt2)^2 \).

\[ (\sqrt5+\sqrt2)^2 =5+2+2\sqrt{10} =7+2\sqrt{10} \] \[ (\sqrt5-\sqrt2)^2 =5+2-2\sqrt{10} =7-2\sqrt{10} \] \[ (7+2\sqrt{10})+(7-2\sqrt{10}) =14 \]

Answer: \(14\)

Exercise 4.3 – Part 4A

Q.5 Simplify by rationalizing the denominator.

(a) Simplify \( \dfrac{2}{\sqrt3} \).

\[ \frac{2}{\sqrt3}\times\frac{\sqrt3}{\sqrt3} =\frac{2\sqrt3}{3} \]

Answer: \(\frac{2\sqrt3}{3}\)

(b) Simplify \( \dfrac{7\sqrt2}{\sqrt5} \).

\[ \frac{7\sqrt2}{\sqrt5}\times\frac{\sqrt5}{\sqrt5} =\frac{7\sqrt{10}}{5} \]

Answer: \(\frac{7\sqrt{10}}{5}\)

(c) Simplify \( \dfrac{9}{\sqrt{10}} \).

\[ \frac{9}{\sqrt{10}}\times\frac{\sqrt{10}}{\sqrt{10}} =\frac{9\sqrt{10}}{10} \]

Answer: \(\frac{9\sqrt{10}}{10}\)

(d) Simplify \( \dfrac{5-\sqrt2}{\sqrt7} \).

\[ \frac{5-\sqrt2}{\sqrt7}\times\frac{\sqrt7}{\sqrt7} =\frac{5\sqrt7-\sqrt{14}}{7} \]

Answer: \(\frac{5\sqrt7-\sqrt{14}}{7}\)

(e) Simplify \( \dfrac{2-\sqrt3}{\sqrt{75}} \).

\[ \sqrt{75}=5\sqrt3 \] \[ \frac{2-\sqrt3}{5\sqrt3}\times\frac{\sqrt3}{\sqrt3} =\frac{2\sqrt3-3}{15} \]

Answer: \(\frac{2\sqrt3-3}{15}\)

Exercise 4.3 – Part 4B

Q.5 Simplify by rationalizing the denominator.

(f) Simplify \( \dfrac{8}{3\sqrt2} \).

\[ \frac{8}{3\sqrt2}\times\frac{\sqrt2}{\sqrt2} =\frac{8\sqrt2}{6} =\frac{4\sqrt2}{3} \]

Answer: \(\frac{4\sqrt2}{3}\)

(g) Simplify \( \dfrac{\sqrt3+\sqrt2}{\sqrt2} \).

\[ \frac{\sqrt3+\sqrt2}{\sqrt2}\times\frac{\sqrt2}{\sqrt2} =\frac{\sqrt6+2}{2} \]

Answer: \(\frac{\sqrt6+2}{2}\)

(h) Simplify \( \dfrac{7}{\sqrt{10}+\sqrt3} \).

\[ \frac{7}{\sqrt{10}+\sqrt3} \times \frac{\sqrt{10}-\sqrt3}{\sqrt{10}-\sqrt3} \] \[ = \frac{7(\sqrt{10}-\sqrt3)} {10-3} \] \[ = \sqrt{10}-\sqrt3 \]

Answer: \(\sqrt{10}-\sqrt3\)

(i) Simplify \( \dfrac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2} \).

\[ \frac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2} \times \frac{\sqrt3+\sqrt2}{\sqrt3+\sqrt2} \] \[ = \frac{(\sqrt3+\sqrt2)^2} {3-2} \] \[ = 3+2+2\sqrt6 \] \[ = 5+2\sqrt6 \]

Answer: \(5+2\sqrt6\)

(j) Simplify \( \dfrac{\sqrt5-\sqrt6}{\sqrt5+\sqrt6} \).

\[ \frac{\sqrt5-\sqrt6}{\sqrt5+\sqrt6} \times \frac{\sqrt5-\sqrt6}{\sqrt5-\sqrt6} \] \[ = \frac{(\sqrt5-\sqrt6)^2} {5-6} \] \[ = \frac{5+6-2\sqrt{30}} {-1} \] \[ = 2\sqrt{30}-11 \]

Answer: \(2\sqrt{30}-11\)

Exercise 4.3 – Part 5

Q.6 If \( \sqrt2 = 1.414 \) and \( \sqrt7 = 2.646 \), find the value of each of the following correct to 2 decimal places.

(a) Find \( \dfrac{3}{\sqrt2} \).

\[ \frac{3}{\sqrt2} =\frac{3}{1.414} \approx2.1216 \] \[ \boxed{2.12} \]

Answer: \(2.12\)

(b) Find \( \dfrac{\sqrt2}{3} \).

\[ \frac{\sqrt2}{3} =\frac{1.414}{3} \approx0.4713 \] \[ \boxed{0.47} \]

Answer: \(0.47\)

(c) Find \( \dfrac{6}{\sqrt7-1} \).

\[ =\frac{6}{2.646-1} =\frac{6}{1.646} \approx3.6452 \] \[ \boxed{3.65} \]

Answer: \(3.65\)

(d) Find \( \dfrac{5+\sqrt2}{5-\sqrt2} \).

\[ =\frac{5+1.414}{5-1.414} =\frac{6.414}{3.586} \approx1.7881 \] \[ \boxed{1.79} \]

Answer: \(1.79\)

(e) Find \( \dfrac{1-\sqrt7}{1+\sqrt7} \).

\[ =\frac{1-2.646}{1+2.646} =\frac{-1.646}{3.646} \approx-0.4515 \] \[ \boxed{-0.45} \]

Answer: \(-0.45\)

Note: All answers have been rounded to 2 decimal places.