- Simplify the following using laws of indices:
(i) (16)-0.75 x (64)4/3
(16)-0.75 x (64)4/3 [16 = 24 ]
= (24)-0.75 x (24)4/3 [64 = 26 ]
= 24 x -3/4 x 26 x 4/3
= 2-3 x 28
= 25
= 32
(ii) (0.25)0.5 x (100)-1/2
(0.25)0.5 x (100)-1/2
= (0.25)1/2 x (1/100 )1/2
= (0.25)1/2 x (1/102 )1/2
= (0.5) x ( 1/10 )
= ( 5/10 ) x ( 1/10)
= 5/100
= 𝟏/𝟐𝟎
(iii) (6.25)0.5 x 102 x (100)-1/2 x (0.01)-1
= ( 625/100 )1/2 x 102 x (1/10)1/2 x ( 1/100 )-1
= ( 25/10 )1/2 x 102 x (1/10 )1/2 x (1/100)-1
= ( 252/10 )1/2 x 102 x (1/102 )1/2 x (100)1
= ( 25/10 ) x 100 x 1 10 x 100
= 2500
(iv) (3-1/2 x 2-1/3) ÷ (3-3/4 x 2-5/6)
= 3−1/2× 2−1/3 3−3/4× 2−5/6
= 3-1/2+3/4 x 2-1/3+5/6
= 31/4 x 23/6
= 31/4 x 21/2
= 31/4 x 21/4
=(3 × 22)1/4
= (3 × 4)1/4
= 𝟏𝟐1/4
- Find the value of the expression.
[ 31/3 {5-1/2 x 3-1/3 x (2252)1/3}1/2]6
Solution:
= [ 31/3 {5-1/2x-1/2 x 3-1/3x1/2 x (2252)2/3x1/2]6
= [ 31/3 {5-1/4 x 31/6 x (2252)-2/6]6
= [ 31/3 x 6 {51/4 x 6 x 31/6 x 6 x (2252)-2/6 x 6]
= [ 32 x 51/4 x 6 x 31/6 x 6 x 2252-2/6 x 6]
= [32 x 53/2 x 31 x 225-2]
= [32 x 52/3 x 31 x 15-4]
= [32 x 5-2/3 x 31 x 3-4 x 5-4]
= [32 + 1 – 4 x 53/2 – 4]
= 1/31 x 1/55/2
= 1/31 x 1/55/2
= 1/31 x 1/√55
= 1/3 x 1/√3125
= 1/3√3125
- Simplify:
[{(35/2 x 53/4) ÷ 2-5/4} ÷ {16 / (52 x 21/4 x 31/2)}]1/5
Solution:
= [{(35/2 × 53/4) ÷ 2−5/4} ÷ {16÷ (52× 21/4× 31/2)}] 1/5
= [{(35/2 × 53/4)/(2−54) ÷ 16/(52× 21/4× 31/2)}] 1/5
= [(35/2 × 53/4)/(2−54) ÷ (52× 21/4× 31/2)/24]1/5
= [(35/2 × 53/4 × 52/1 × 21/4 × 31/2)/(2−5/4 × 24 )]1/5
= [21/4 × 35/2 + 1/2 × 53/4 + 2/1/2−5/4 + 4/1]1/5
= 21/4 × 35/2 + 1/2 × 511/4/211/4]1/5
= [21/4 – 4/11 × 33 × 511/4 ] 1/5
= [210/4 × 1/5 × 33 × 1/5 × 511/4 + 1/5]
= 2−1/2 × 31/5 × 511/20
