Fsc Part 1 Mathematics (Complete Solution)

# Q7 Simplify the following

(i) ( -
 1 2
+
 √  3  ι 2
) 3

SOLUTION:

= (
 -1 + √  3  ι 2
) 3

=
 (-1+√  3  ι)3 (2)3

=
 (-1)3 + (√  3 ι)3+3(-1)(√  3  ι)(-1+√  3 ι) 8

=
 -1+3√  3 ι3-3√ 3  ι(-1+√  3 ι) 8

=
 -1+3√  3 ι2.ι +3√ 3  ι-3ι2√  (3)(3) 8

=
 -1+3√  3 (-1).ι +3√ 3  ι-3(-1)(3) 8

=
 -1-3√  3 ι +3√ 3  ι+9 8

=
 8 8

=1

(ii) ( -
 1 2
-
 √  3  ι 2
) 3

SOLUTION:

= (
 -1-√  3  ι 2
) 3

=
 (-1-√  3  ι)3 (2)3

=
 (-1)3+(-√  3  ι)3+3(-1)(-√  3  ι)(-1-√  3  ι) 8

=
 -1-3√  3 ι3+3√ 3  ι(-1-√  3 ι) 8

=
 -1-3√  3 ι2.ι -3√ 3  ι-3ι2√  (3)(3) 8

=
 -1-3√  3 (-1).ι -3√ 3  ι-3(-1)(3) 8

=
 -1+3√  3 ι -3√ 3  ι+9 8

=
 8 8

=1

(iii) ( -
 1 2
-
 √  3  ι 2
) - 2 + 1

SOLUTION:

= (
 -1-√  3  ι 2
) - 1

=
2
 -1-√  3  ι

= -
2
 1+√  3  ι

= -
2
 1+√  3  ι
x
1-√  3  ι
 1-√  3  ι

= -
2(1-√  3  ι)
 (1+√  3  ι)(1-√  3  ι)

=
-2(1-√  3  ι)
 (1)2-(√  3  ι)2

=
 -2(1-√  3  ι) 1-3ι2

=
 -2(1-√  3  ι) 1-3(-1)

=
 -2(1-√  3  ι) 4

=
 -1+√  3  ι) 2

(iv) (a+bι)2

Solution:

=(a)2+(bι)2+2(a)(bι)

=a2+b2ι2+2abι

=a2+b2(-1)+2abι

=a2-b2+2abι

(v) (a+bι)-2

SOLUTION:(a+bι)-2

=[(a+bι)2]-1

=[(a)2+(bι)2+2(a)(bι)]-1

=(a2+b2ι2+2abι)-1

=[a2+b2(-1)+2abι]-1

=(a2-b2+2abι)-1

=
 1 (a2-b2+2abι)
x
 (a2-b2-2abι) (a2-b2-2abι)

=
 1(a2-b2-2abι) (a2-b2+2abι)(a2-b2-2abι)

=
 a2-b2-2abι (a2-b2)2-(2abι)2

=
 a2-b2-2abι (a4+b4-2a2b2)-4a2b2ι2

=
 a2-b2-2abι a4+b4-2a2b2-4a2b2(-1)

=
 a2-b2-2abι a4+b4-2a2b2+4a2b2

=
 a2-b2-2abι a4+b4+2a2b2

=
 (a2-b2)-2abι (a2+b2)2

=
 a2-b2 (a2+b2)2
-
 2abι (a2+b2)2

(vi) (a+bι)3

SOLUTION:(a+bι)3

=(a)3+(bι)3+3(a)(bι)(a+bι)

=a3+b3ι3+3abι(a+bι)

=a3+b3ι3+3abι(a+bι)

=a3+b3ι2.ι+3a2bι+3ab2ι2

=a3+b3ι(-1)+3a2bι+3ab2(-1)

=a3-b3ι+3a2bι-3ab2

=a3-3ab2 +(3a2b-b3

(vii) (a-bι)3

SOLUTION:(a-bι)3

=(a)3+(-bι)3+3(a)(-bι)(a-bι)

=a3-b3ι3-3abι(a-bι)

=a3-b3ι2.ι-3a2bι+3ab2ι2

=a3-b3ι(-1)-3a2bι+3ab2(-1)

=a3+b3ι-3a2bι-3ab2

=a3-3ab2 -(3a2b-b3

 (viii) (3 - √ -4 )-3

 SOLUTION: (3 - √ -4 )-3

 = [(3 - √ -4 )3]-1

= [(3-2ι)3]-1

= [(3)3+(-2ι)3+3(3)(-2ι)(3-2ι)]-1

= [27-8ι3-18ι)(3-2ι)]-1

= [27-8ι2.ι-54ι+36ι2]-1

= [27-8(-1).ι-54ι+36(-1)]-1

= [27+8ι-54ι-36]-1

= [-9-46ι]-1

=
 1 -9-46ι
x
 -9+46ι -9+46ι

=
 1(-9+46ι) (-9-46ι)(-9+46ι)

=
 -9+46ι (-9)2-(46ι)2

=
 -9+46ι 81-2116ι2

=
 -9+46ι 81-2116(-1)

=
 -9+46ι 2197

= -
 9 2197
+
 46ι 2197