Find [2^-1+3^-1+4^0]^-1
Answers
Answer:
i) (30 + 4-1) * 22 = (1 + 1/4) * 22 [a0 = 1 and a-m = 1/ am]
= (5/4) * 22
= (5/22) * 22
= 5 * 22-2 [am ÷ an = am-n]
= 5 * 20
= 5 * 1
= 5
(ii) (2-1 * 4-1) ÷ 2-2 = (1/2 * 1/4) ÷ 2-2 [a-m = 1/ am]
= (1/8) ÷ 2-2
= (1/8) ÷ 2-2
= (1/23) ÷ 2-2
= 2-3 ÷ 2-2
= 2-3 * 1/2-2
= 2-3+2 [am ÷ an = am-n]
= 2-1
= 1/2 [a-m = 1/ am]
(iii) (1/2)-2 + (1/3)-2 + (1/4)-2 = (2/1)2 + (3/1)2 + (4/1)2 [a-m = 1/ am]
= 22 + 32 + 42
= 4 + 9 + 16
= 29
(iv) (3-1 + 4-1 + 5-1)0 = 1 [(a + b)0 = 1]
(v) {(-2/3)-2}2 = {(-3/2)2}2 [a-m = 1/ am]
= (-3/2)2*2 [(am)n = am*n]
= (-3/2)4
= (-3)4/24
= 81/16
Question 4:
Evaluate:
(i) (8-1 * 53)/2-4 (ii) (5-1 * 2-1) * 6-1
Answer:
(i) (8-1 * 53)/2-4 = {(23)-1 * 53}/2-4
= (2-3 * 53)/2-4 [(am)n = am*n]
= (2-3+4 * 53) [am ÷ an = am-n]
= 2 * 53
= 2 * 125
= 250
(ii) (5-1 * 2-1) * 6-1 = (1/5 * 1/2) * 1/6 [a-m = 1/ am]
= 1/10 * 1/6
= 1/60
Question 5:
Find the value of m for which 5m ÷ 5-3 = 55
Answer:
Given, 5m ÷ 5-3 = 55
=> 5m-(-3) = 53 [am ÷ an = am-n]
=> 5m+3 = 55
Comparing exponent on both sides, we get
=> m + 3 = 5
=> m = 5 -3
=> m = 2
Question 6:
Evaluate:
(i) {(1/3)-1 + (1/4)-1}-1 (ii) (5/8)-7 * (8/5)-4
Answer:
(i) {(1/3)-1 - (1/4)-1}-1 = {(3/1)1 - (4/1)1}-1 [a-m = 1/ am]
= (3 - 4)-1
= (-1)-1
= 1/(-1) [a-m = 1/ am]
= -1
(ii) (5/8)-7 * (8/5)-4 = 5-7/8-7 * 8-4/5-4 [a-m = 1/ am]
= 5-7+4 * 8-4-(-7) [am ÷ an = am-n]
= 5-3 * 83
= 83/53 [a-m = 1/ am]
= 512/125
Question 7:
Simplify:
(i) (25 * t-4)/(5-3 * 10 * t-8) (t ≠ 0) (ii) (3-5 * 10-5 * 125)/(5-7 * 6-5)
Answer:
(i) (25 * t-4)/(5-3 * 10 * t-8) = (52 * t-4)/(5-3 * 2 * 5 * t-8) [am ÷ an = am-n]
= (52 * t-4-(-8))/(5-3+1 * 2)
= (52 * t-4+8)/(5-2 * 2)
= (52-(-2) * t4)/2 [am ÷ an = am-n]
= (52+2 * t4)/2
= (54 * t4)/2
= 625t4/2
(ii) (3-5 * 10-5 * 125)/(5-7 * 6-5) = {3-5 * (2 * 5)-5 * 53}/(5-7 * 6-5)
= {3-5 * 2-5 * 5-5 * 53}/{5-7 * (2 * 3)-5}
= {3-5 * 2-5 * 5-5 * 53}/{5-7 * 2-5 * 3-5}
= 3-5-(-5) * 2-5-(5) * 5-5+3-(-7) [am ÷ an = am-n]
= 3-5+5 * 2-5+5 * 5-5+3 +7
= 30 * 20 * 5-5+10
= 1 * 1 * 55 [a0 = 1]
= 3125
Exercise 12.2
Question 1:
Express the following numbers in standard form:
(i) 0.0000000000085 (ii) 0.00000000000942 (iii) 6020000000000000
(iv) 0.00000000837 (v) 31860000000
Answer:
(i) 0.0000000000085 = 0.0000000000085 * 1012/1012 = 8.5/1012 = 8.5* 10-12
(ii) 0.00000000000942 = 0.00000000000942 * 1012/1012 = 9.42/1012 = 9.42* 10-12
(iii) 6020000000000000 = 602 * 1015
(iv) 0.00000000837 = 0.00000000837 * 109/109 = 8.37/109 = 8.37* 10-9
(v) 31860000000 = 3.186 * 1010