Exponents and powers are the basics of mathematics. Various exponent formulae are used in the various fields of mathematics. This article is about the evaluation of (√4)-3. The exponents and powers can be evaluated very easily. Exponents are very useful in various formulas and mathematics. Exponent is the power on the variable or any constant when it is multiplied n times.
Exponents
When a number, constant, or variable (a) is multiplied p times, then it results in ap, then p is called as exponent of a. An exponent is the power on any number, constant or variables. Example: a × a × a × a…p times = ap where a is called the base and p is called the exponent.
Evaluate (√4)-3
Solution:
For evaluation of (√4)-3 following exponent formulae are used,
a-p = 1/ap
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a1/p = p√a
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(ap)q = apq
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a.a.a….. p times = ap
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Let r = (√4)-3
Now, we have to evaluate p using exponents formulae,
r = (√4)-3
r = 1/(√4)3 [(√4)-3 = 1/ (√4)3]
r = 1/[(4)1/2]3 [√4 = 41/2]
r = 1/[(22)1/2]3 [4 = 22]
r = 1/22×(1/2)×3 [(22)1/2]3 = 22×(1/2)×3
r = 1/23
r = 1/8 [23 = 2 × 2 × 2 = 8]
Therefore, the evaluation of (√4)-3 = 1/8
Sample Problems
Question 1: Evaluate:
Solution:
- 5.5.5.5 = 54 =625
- (32)2 = 32×2 = 34 = 81
- 45/2 = (22)5/2 = 22×(5/2) = 25 = 32
Question 2: Simplify: ( 2-2×72) / 5-1
Solution:
(2-2 × 72) / 5-1 = (51 × 72) / 22
= (5 × 49)/4
= 245/4
Question 3: Find the value of y: (8)y+3 = 4y.26
Solution:
(8)y+3 = 4y.26
[(2)3]y+3 = [(2)2]y.26
23(y+3) = 22×y . 26
23y+9 = 22y+6
Since, bases are same equate powers
3y+ 9 = 2y+6
3y – 2y = 6 – 9
y = -3
Question 4: Find the value of (7292/3)1/2
Solution:
(7292/3)1/2 = {(93)2/3}1/2
= (9)3×(2/3)×(1/2)
= 91
(7292/3)1/2 = 9
Question 5: Find the value of a × b if,
4a = 2a+5; 25b-2 = 5b+4
Solution:
4a = 2a+5
(22)a = 2a+5
22a = 2a+5
Since, bases are same equate the powers
2a = a + 5
a = 5
25b-2 = 5b+4
(52)b-2 = 5b+4
52(b-2) = 5b+4
52b-4 = 5b+4
Since, bases are same equate the powers
2b – 4 = b + 4
2b – b = 4 + 4
b = 8
a × b = 5 × 8 = 40
Question 6: If (-6)p.(36)p-2 = (-6)5, then find (p2 + 4) / (p2 – 4).
Solution:
(-6)p.(36)p-2 = (-6)5
(-6)p.{(-6)2}p-2 = (-6)5
(-6)p.(-6)2(p-2) = (-6)5
(-6)p.(-6)2p-4 = (-6)5
(-6)p+2p-4 = (-6)5
(-6)3p-4 = (-6)5
Since, bases are same equate the powers,
3p – 4 = 5
3p = 9
p = 3
(p2 + 4) / (p2 – 4) = (32 + 4) / (32 – 4)
= (9 + 4) / (9 – 4)
(p2 + 4) / (p2 – 4) = 13 / 5
Question 7: Find the multiplicative inverse of [(14)-1]2÷(84)-2.
Solution:
[(14)-1]2 ÷ (84)-2 = (14)-2 ÷ (1/842)
= (1/142) × 842
= 842 / 142
= (84 × 84)/(14 × 14)
= 6 × 6
[(14)-1]2 ÷ (84)-2 = 36
Last Updated :
30 Dec, 2023
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