Here is a list of solutions from Algebra chapter form Vedanta Excel in Mathematics Book 10:

Exercise 11.1

General Section

1. Simplify:

b) 5√3 + 6√3 - √3

Solution:
Given,

= 5√3 + 6√3 - √3

= (5+6-1)√3

= 10√3

d) 8\sqrt[3]{4} - \sqrt[3]{4} - 3\sqrt[3]{4}

Solution:
Given,

= 8\sqrt[3]{4} - \sqrt[3]{4} - 3\sqrt[3]{4}

= (8-1-3) \sqrt[3]{4}

= 4 \sqrt[3]{4}


f) 2\sqrt[4]{6} - \sqrt[4]{6} - 3\sqrt[4]{6}

Solution:
Given,

= 2\sqrt[4]{6} - \sqrt[4]{6} - 3\sqrt[4]{6}

= (2-1-3) \sqrt[4]{6}

= -2 \sqrt[4]{6}

2. Simplify:

b) √6 × √3 × 2√2

Solution:
Given,

= √6 × √3 × 2√2

= 2 \sqrt{6 × 3 × 2}

= 2 \sqrt{36}

= 2 \sqrt{6^2}

= 2 × 6

= 12


c) \sqrt[3]{9} × \sqrt[3]{3} × 3\sqrt[3]{2}

Solution:
Given,

= \sqrt[3]{9} × \sqrt[3]{3} × 3\sqrt[3]{2}

= 3 \sqrt[3]{9×3×2}

= 3 \sqrt[3]{27×2}

= 3 \sqrt[3]{3^3 × 2}

= 3× 3\sqrt[3]{2}

= 9 \sqrt[3]{2}


e) 5 \sqrt[3]{108} ÷ 3\sqrt[3]{2}

Solution:
Given,

= 5\sqrt[3]{108} ÷ 3\sqrt[3]{2}

= \dfrac{5\sqrt[3]{108}}{3\sqrt[3]{2}}

= \dfrac{5}{3} × \sqrt[3]{\dfrac{108}{2}}

= \dfrac{5}{3} × \sqrt[3]{54}

= \dfrac{5}{3} × \sqrt[3]{3^3×2}

= \dfrac{5}{3} × 3 × \sqrt[3]{2}

= 5 \sqrt[3]{2}

3. Simplify:

a) √32 + √8 - √72

Solution:
Given,

= √32 + √8 - √72

= \sqrt{16 × 2} + \sqrt{4 × 2} - \sqrt{36 × 2}

= \sqrt{4^2 × 2} + \sqrt{2^2 × 2} - \sqrt{6^2 × 2}

= 4 √2 + 2√2 - 6√2

= √2 ( 4+2-6)

= √2 (0)

= 0


b) √27 + √75 - 8√3

Solution:
Given,

= √27 + √75 - 8√3

= \sqrt{3^2 × 3} + \sqrt{5^2 × 3} - 8√3

= 3 √3 + 5√3 - 8√3

= (3+5-8)√3

= (0)√3

= 0



d) √12 - √75 + √48

Solution:
Given,

= √12 - √75 + √48

= \sqrt{2^2 × 3} - \sqrt{5^2×3} + \sqrt{4^2 × 3}

= 2√3 - 5√3 + 4√3

= (2-5+4)√3

= √3


f) 5\sqrt[3]{81} - 2\sqrt[3]{24} + \sqrt[3]{375}

Solution:
Given,

= 5\sqrt[3]{81} - 2\sqrt[3]{24} + \sqrt[3]{375}

= 5 \sqrt[3]{3^3 × 3} - 2\sqrt[3]{2^3×3} + \sqrt[3]{5^3×3}

= 5×3 \sqrt[3]{3} - 2×2\sqrt[3]{3} + 5\sqrt[3]{3}

= 15\sqrt[3]{3} - 4\sqrt[3]{3} + 5\sqrt[3]{3}

= (15-4+5)\sqrt[3]{3}

= 16\sqrt[3]{3}


h) 3√2 + \sqrt[4]{2500} - \sqrt[4]{64} + 6√8

Solution:
Given,

= 3√2 + \sqrt[4]{2500} - \sqrt[4]{64} + 6√8

= 3√2 + \sqrt[4]{5^4 × 4} - \sqrt[4]{2^4×4} + 6\sqrt{2^2×2}

= 3√2 + 6\sqrt{2^2 × 2} + \sqrt[4]{5^4×4} - \sqrt[4]{2^4 ×4}

= 3√2 + 6×2√2 + 5\sqrt[4]{4} - 2\sqrt[4]{4}

= (3+12)√2 + (5-2)\sqrt[4]{4}

= 15√2 + 3\sqrt[4]{4}

4. Simplify:


a) (√3 +√2)(√3 -√2)

Solution:
Given,

= (√3+√2)(√3-√2)

= (√3)² - (√2)²

= 3 - 2

= 1


b) (√5 - √3) (√5 + √3)

Solution:
Given,

= (√5 - √3) (√5 + √3)

= (√5)² - (√3)²

= 5 -3

= 2


c) (2√5 + 3√2) (2√5 - 3√2)

Solution:
Given,

= (2√5 + 3√2) (2√5 - 3√2)

= (2√5)² - (3√2)²

= 4×5 - 9×2

= 20 -18

= 2


d) (√2 + √3)²

Solution:
Given,

= (√2 + √3)²

= (√2)² + 2×√2×√3 + (√3)²

= 2 + 2\sqrt{2×3} + 3

= 5 + 2√6


e) (√5 - √3)²

Solution:
Given,

= (√5 - √3)²

= (√5)² - 2×√5×√3 + (√3)²

= 5 - 2\sqrt{5×3} + 3

= 8 - 2\sqrt{15}

= 2(4 - \sqrt{15})


f) \left ( \sqrt{x+a} - \sqrt{x-a} \right )²

Solution:
Given,

= \left ( \sqrt{x+a} - \sqrt{x-a} \right )²

= (\sqrt{x+a} )² - 2 × \sqrt{x+a} × \sqrt{x-a} + ( \sqrt{x-a} )²

= (x+a) - 2 × \sqrt{(x+a)(x-a)} + (x-a)

= x+a+x-a - 2\sqrt{x²-a²}

= 2x - 2\sqrt{x²-a²}

= 2(x -\sqrt{x²-a²}


g) (2√2 - √3) (3√2 + √3)

Solution:
Given,

= (2√2 - √3) (3√2 +√3)

= 2√2 (3√2 + √3) - √3(3√2+√3)

= 6(√2×√2) + 2(√2×√3) -3(√2×√3) - (√3×√3)

= 6×2 + 2√6 - 3√6 -3

= 12 -3 +(2-3)√6

= 9 - √6


h) (3√5 - 4√2) (2√5 + 2√3)

Solution:
Given,

=  (3√5 - 4√2) (2√5 + 2√3)

= 3√5 (2√5 +2√3) - 4√2(2√5 +2√3)

= 6(√5×√5) + 6(√5×√3) -8(√2×√5) + 8(√2×√3)

= 6×5 + 6\sqrt{15} - 8\sqrt{10} + 8√6

= 30 + 6\sqrt{15} - 8\sqrt{10} + 8√6

5. Simplify:

b) \dfrac{\sqrt{x²-9}}{\sqrt{x-3}}

Solution:
Given,

= \dfrac{\sqrt{x²-9}}{\sqrt{x-3}}

= \dfrac{\sqrt{(x)²-(3)³}}{\sqrt{x-3}}

= \dfrac{\sqrt{(x+3)(x-3)}}{\sqrt{x-3}}

= \dfrac{\sqrt{x+3} × \sqrt{x-3}}{\sqrt{x-3}}

= \sqrt{x+3}



d) \dfrac{x-4}{√x+2}

Solution:
Given,

= \dfrac{x-4}{√x +2}

= \dfrac{(√x)² -2²}{√x +2}

= \dfrac{(√x -2) (√x +2)}{√x +2}

= √x -2


f) \dfrac{49-5x}{7-\sqrt{5x}}

Solution:
Given,

= \dfrac{49-5x}{7- \sqrt{5x}}

= \dfrac{(7)² - (\sqrt{5x})²}{7 - \sqrt{5x}}

= \dfrac{(7 - \sqrt{5x}) (7+ \sqrt{5x)}}{7 - \sqrt{5x}}

= 7 + \sqrt{5x}



6. Simplify:

b) \dfrac{4 \sqrt[3]{54} - 2\sqrt[3]{250}}{6\sqrt[3]{128}}

Solution:
Given,

= \dfrac{4 \sqrt[3]{54} - 2\sqrt[3]{250}}{6\sqrt[3]{128}}

= \dfrac{4\sqrt[3]{3^3 × 2} - 2\sqrt[3]{5^3 × 2}}{6\sqrt[3]{4^3 × 2}}

= \dfrac{4×3\sqrt[3]{2} - 2×5\sqrt[3]{2}}{6×4\sqrt[3]{2}}

= \dfrac{12\sqrt[3]{2} - 10\sqrt[3]{2}}{24\sqrt[3]{2}}

= \dfrac{(12-10)\sqrt[3]{2}}{24\sqrt[3]{2}}

= \dfrac{2\sqrt[3]{2}}{24 \sqrt[3]{2}}

= \dfrac{2}{24} × \sqrt[3]{\dfrac{2}{2}}

= \dfrac{1}{12} × \sqrt[3]{1}

= \dfrac{1}{12}



d) \dfrac{3\sqrt[3]{81} - 3\sqrt[3]{24} + 2\sqrt[3]{275}}{13 \sqrt[3]{192}}

Solution:
Given,

= \dfrac{3\sqrt[3]{81} - 3\sqrt[3]{24} + 2\sqrt[3]{275}}{13 \sqrt[3]{192}}

= \dfrac{3\sqrt[3]{3^3 × 3} - 3\sqrt[3]{2^3 × 3} + 2\sqrt[3]{5^3×3}}{13 \sqrt[3]{4^3×3}}

= \dfrac{3×3 \sqrt[3]{3} - 3×2 \sqrt[3]{3} + 2×5 \sqrt[3]{3}}{13×4 \sqrt[3]{3}}

= \dfrac{9 \sqrt[3]{3} - 6\sqrt[3]{3} + 10\sqrt[3]{3}}{52\sqrt[3]{3}}

= \dfrac{ (9-6+10) \sqrt[3]{3}}{52 \sqrt[3]{3}}

= \dfrac{13 \sqrt[3]{3} }{52 \sqrt[3]{3}}

= \dfrac{13}{52} × \sqrt[3]{\dfrac{3}{3}}

= \dfrac{1}{4} × 1

= \dfrac{1}{4}

f) \dfrac{5 \sqrt[3]{81} - 2\sqrt[3]{24}}{2 \sqrt[3]{48} + 3\sqrt[3]{162}}

Solution:
Given,

= \dfrac{5 \sqrt[3]{81} - 2\sqrt[3]{24}}{2 \sqrt[3]{48} + 3\sqrt[3]{162}}

= \dfrac{5 \sqrt[3]{3^3 × 3} - 2\sqrt[3]{2^3 × 3}}{2 \sqrt[3]{2^3 × 6} + 3\sqrt[3]{3^3×6}}

= \dfrac{5 × 3 \sqrt[3]{3} - 2×2 \sqrt[3]{3}}{2×2 \sqrt[3]{6} + 3×3 \sqrt[3]{6}}

= \dfrac{15 \sqrt[3]{3} - 4\sqrt[3]{3}}{4 \sqrt[3]{6} + 9\sqrt[3]{6}}

= \dfrac{(15-4) \sqrt[3]{3}}{(4+9)\sqrt[3]{6}}

= \dfrac{11 \sqrt[3]{3}}{13 \sqrt[3]{6}}

= \dfrac{11}{13} × \sqrt[3]{\dfrac{3}{6}}

= \dfrac{11}{13} × \sqrt[3]{\dfrac{1}{2}}

= \dfrac{11 × \sqrt[3]{1}}{13 × \sqrt[3]{2}}

= \dfrac{11}{13 \sqrt[3]{2}}

Class 10 - Mathematics - vedanta - Algebra - Indices - Simplification Of Surds - Exercise 11.1 - Solutions - Solved Exercises

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