StudentAidUnit 1: Real Numbers — Exercise 1 2
9th Class Mathematics · Unit 1: Real Numbers
1.2.1.Rationalize the denominator of the following: (i) \frac{13}{4+\sqrt{3}} (ii) \frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}} (iii) \frac{\sqrt{2}-1}{\sqrt{5}} (iv) \frac{6-4\sqrt{2}}{6+4\sqrt{2}} (v) \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}} (vi) \frac{4\sqrt{3}}{\sqrt{7}+\sqrt{5}}
(i)
134+sqrt3 × 4-sqrt34-sqrt3=13(4-sqrt3)16-3=4-sqrt3
(ii)
sqrt2+sqrt5sqrt3 × sqrt3sqrt3=frac{sqrt6+sqrt{15}{3}
(iii)
sqrt2-1sqrt5 × sqrt5sqrt5=frac{sqrt{10}-sqrt5}{5}
(iv)
(6-4sqrt2)236-32=68-48sqrt24=17-12sqrt2
(v)
(sqrt3-sqrt2)23-2=5-2sqrt6
(vi)
4sqrt3(sqrt7-sqrt5)7-5=2sqrt3(sqrt7-sqrt5)
(i) 4-sqrt3 (ii) frac{sqrt6+sqrt{15}{3} (iii) frac{sqrt{10}-sqrt5}{5} (iv) 17-12sqrt2 (v) 5-2sqrt6 (vi) 2sqrt3(sqrt7-sqrt5)
1.2.2.Simplify the following: (i) \left(\frac{81}{16}\right)^{-3/4} (ii) \left(\frac{3}{4}\right)^{-2}\div\left(\frac{4}{9}\right)^{3}\times\frac{16}{27} (iii) (0.027)^{-1/3} (iv) \sqrt[7]{\frac{x^{14}\times y^{21}\times z^{35}}{y^{14}z^{7}}} (v) \frac{5\cdot(25)^{n+1}-25\cdot(5)^{2n}}{5\cdot(5)^{2n+3}-(25)^{n+1}} (vi) \frac{(16)^{x+1}+20(4^{2x})}{2^{x-3}\times8^{x+2}} (vii) (64)^{-2/3}\div(9)^{-3/2} (viii) \frac{3^n\times9^{n+1}}{3^{n-1}\times9^{n-1}} (ix) \frac{5^{n+3}-6\cdot5^{n+1}}{9\times5^n-4\times5^n}
Result
(i)
left(8116right)-3/4=left(1681right)3/4=2333=827
(ii)
left(43right)2divleft(49right)3 × 1627=16 × 729 × 169 × 64 × 27=12
(iii)
(0.027)-1/3=left(100027right)1/3=103
(iv)
(x14y7z28)1/7=x2yz4
(v)
52n+2(5-1)52n+2(52-1)=424=16
(vi)
frac{24x(16+20)}{24x+3=368=92
(vii)
(64)-2/3div(9)-3/2=4-2 × 33=2716
(viii)
frac{33n+2{33n-3=35=243
(ix)
5n(125-30)5n(9-4)=955=19
1.2.3.If x=3+\sqrt8 then find the value of: (i) x+\frac{1}{x} (ii) x-\frac{1}{x} (iii) x^2+\frac{1}{x^2} (iv) x^2-\frac{1}{x^2} (v) x^4+\frac{1}{x^4} (vi) \left(x-\frac{1}{x}\right)^2
Given
x=3+sqrt8 ⇒ 1x=3-sqrt8
Result
(i)
x+frac1x=(3+sqrt8)+(3-sqrt8)=6
(ii)
x-frac1x=(3+sqrt8)-(3-sqrt8)=2sqrt8
(iii)
x2+frac1{x2}=left(x+frac1xright)2-2=36-2=34
(iv)
x2-frac1{x2}=left(x+frac1xright)left(x-frac1xright)=6 × 2sqrt8=12sqrt8
(v)
x4+frac1{x4}=left(x2+frac1{x2}right)2-2=342-2=1154
(vi)
left(x-frac1xright)2=(2sqrt8)2=32
1.2.4.Find the rational numbers p and q such that \frac{8-3\sqrt2}{4+3\sqrt2}=p+q\sqrt2
Given
8-3sqrt24+3sqrt2=p+qsqrt2
8-3sqrt24+3sqrt2 × 4-3sqrt24-3sqrt2=50-36sqrt216-18=50-36sqrt2-2
Result
-25+18sqrt2=p+qsqrt2 ⇒ p=-25, q=18
1.2.5.Simplify the following: (i) \frac{(25)^{3/2}\times(243)^{3/5}}{(16)^{5/4}\times(8)^{4/3}} (ii) \frac{54\times\sqrt[3]{(27)^{2x}}}{9^{x+1}+216(3^{2x-1})} (iii) \sqrt{\frac{(216)^{2/3}\times(25)^{1/2}}{(0.04)^{-3/2}}} (iv) \left(a^{1/3}+b^{2/3}\right)\times\left(a^{2/3}-a^{1/3}b^{2/3}+b^{4/3}\right)
Result
(i)
53 × 3325 × 24=125 × 27512=3375512
(ii)
frac{54 × 32x{32x(9+72)}=5481=frac23
(iii)
sqrt{36 × 5125=sqrt{3625=frac65
(iv)
a+b2 (sum of cubes: x3+y3 with x=a1/3,y=b2/3)