Exercise 4.1 – Exponents
1. Write the base and exponent of each of the following
(a)
24
Base = 2
Exponent = 4
Exponent = 4
(b)
(−3)5
Base = −3
Exponent = 5
Exponent = 5
(c)
(34)5
Base =
34
Exponent = 5
Exponent = 5
(d)
1
54
Rewriting:
154
=
154
Base = 5, exponent = −4
Base = 5, exponent = −4
(e)
(−23)3
Base =
−23
Exponent = 3
Exponent = 3
2. Simplify
(a)
32 ×
135
× (32)6
= 32 × 3−5 × 312
= 32 − 5 + 12
= 39
= 19683
= 32 − 5 + 12
= 39
= 19683
(b)
(−5)3 × (−5)4 × (−5)2
= (−5)3 + 4 + 2
= (−5)9
= −1953125
= (−5)9
= −1953125
(c)
(25)2
×
(25)−3
×
(25)4
=
(25)2 − 3 + 4
= (25)3
= 8125
= (25)3
= 8125
(d)
[(37)−1]−2
= (37)(−1)(−2)
= (37)2
= 949
= (37)2
= 949
3. Find the value of x
(a)
5x − 2 = 25
25 = 52
Therefore, 5x − 2 = 52
x − 2 = 2
x = 4
Therefore, 5x − 2 = 52
x − 2 = 2
x = 4
(b)
(22)x = (23)4
22x = 212
2x = 12
x = 6
2x = 12
x = 6
(c)
2x + 2x + 2x = 192
3 × 2x = 192
2x = 64
64 = 26
x = 6
2x = 64
64 = 26
x = 6
(d)
8255 = 32x
8 = 23 and 32 = 25
(23)255 = (25)x
2765 = 25x
765 = 5x
x = 153
(23)255 = (25)x
2765 = 25x
765 = 5x
x = 153
(e)
7x × 2x × 3x = 1764
(7 × 2 × 3)x = 1764
42x = 1764
1764 = 422
x = 2
42x = 1764
1764 = 422
x = 2
(f)
4(10x)2
− 6(10x)2
+ 3(10x)2
= 1
Let
y = (10x)2.
Then, 4y − 6y + 3y = 1
y = 1
Therefore, (10x)2 = 1
10x = ±1
10 = ±x
x = ±10
Then, 4y − 6y + 3y = 1
y = 1
Therefore, (10x)2 = 1
10x = ±1
10 = ±x
x = ±10
4. Evaluate the following
(a)
(12)5
×
(−23)4
×
(35)−1
=
132
×
1681
×
53
= 807776
= 5486
= 807776
= 5486
(b)
73 × 49−1 × 53 × 25−4
= 73 × 7−2 × 53 × 5−8
= 71 × 5−5
= 755
= 73125
= 71 × 5−5
= 755
= 73125
(c)
(125)3
×
56144
=
12353
×
56144
= 12 × 53
= 1500
= 12 × 53
= 1500
(d)
93 × 27 × 814
3−2 × 34 × 812
Expressing all numbers as powers of 3:
Numerator: 93 × 27 × 814 = 36 × 33 × 316 = 325
Denominator: 3−2 × 34 × 812 = 3−2 × 34 × 38 = 310
Therefore, 325 310 = 315
= 14348907
Numerator: 93 × 27 × 814 = 36 × 33 × 316 = 325
Denominator: 3−2 × 34 × 812 = 3−2 × 34 × 38 = 310
Therefore, 325 310 = 315
= 14348907
Answer Key
1.
(a) Base 2, exponent 4 |
(b) Base −3, exponent 5 |
(c) Base 3/4, exponent 5 |
(d) Base 5, exponent −4 |
(e) Base −2/3, exponent 3
2.
(a) 19683 |
(b) −1953125 |
(c) 8/125 |
(d) 9/49
3.
(a) x = 4 |
(b) x = 6 |
(c) x = 6 |
(d) x = 153 |
(e) x = 2 |
(f) x = ±10
4.
(a) 5/486 |
(b) 7/3125 |
(c) 1500 |
(d) 14348907