simplify and express in power notation. (81/16)^(-1/2) × (2/3)^2 ÷ (3/2)^(-2)
Answers
Step-by-step explanation:
Evaluate:
(i) 3-2 (ii) (-4)-2 (iii) (1/2)-5
Answer:
(i) 3-2 = 1/32 = 1/9 [a-m = 1/ am]
(ii) (-4)-2 = 1/42 = 1/16 [a-m = 1/ am]
(iii) (1/2)-5 = (2/1)5 = 25 = 32 [a-m = 1/ am]
Question 2:
Simplify and express the result in power notation with positive exponent:
(i) (-4)5 ÷ (-4)8 (ii) (1/23)2 (iii) (-3)4 * (5/3)4 (iv) (3-7 * 3-10) * 35
(v) 2-3 * (-7)3
Answer:
(i) (-4)5 ÷ (-4)8 = (-4)5-8 [am ÷ an = am-n]
= (-4)-3
= 1/(-4)3 [a-m = 1/ am]
= -1/64
(ii) (1/23)2 = 12/(23)2 [(a/b)m = am/bm]
= 1/ 23*2 [(am)n = am*n]
= 1/26
= 1/64
(iii) (-3)4 * (5/3)4 = (-3)4 * (54/34 ) [(a/b)m = am/bm]
= (3)4 * (54/34 ) [(-a)m = am when m is an even number]
= (3)4-4 * 54
= 54
(iv) (3-7 * 3-10) * 35 = 3-7-10+5 [am * an = am+n]
= 3-17+5
= 3-12
= 1/312 [a-m = 1/ am]
(v) 2-3 * (-7)-3 = 1/23 * 1/(-7)-3 [a-m = 1/ am]
= 1/{(-7)3 * 23 }
= 1/(-7 * 2)3 [am * bm = (a * b)m]
= 1/(-14)3
= -1/(14)3 [(-a)m = -am when m is an odd number]
Question 3:
Find the value of:
(i) (30 + 4-1) * 22 (ii) (2-1 * 4-1) ÷ 2-2 (iii) (1/2)-2 + (1/3)-2 + (1/4)-2
(iv) (3-1 + 4-1 + 5-1)0 (v) {(-2/3)-2}2
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