LAWS OF EXPONENTS
About "Laws of Exponents"
Laws of Exponents :
In this section, we are going to see the different laws of exponents.
Laws of Exponents
Law 1 :
xm ⋅ xn = xm+n
Example :
34 ⋅ 35 = 34+5
34 ⋅ 35 = 39
Law 2 :
xm ÷ xn = xm-n
Example :
37 ÷ 35 = 37-5
37 ÷ 35 = 32
Law 3 :
(xm)n = xmn
Example :
(32)4 = 3(2)(4)
(32)4 = 38
Law 4 :
(xy)m = xm ⋅ ym
Example :
(3 ⋅ 5)2 = 32 ⋅ 52
(3 ⋅ 5)2 = 9 ⋅ 25
(3 ⋅ 5)2 = 225.
Law 5 :
(x / y)m = xm / ym
Example :
(3 / 5)2 = 32 / 52
(3 / 5)2 = 9 / 25
Law 6 :
x-m = 1 / xm
Example :
3-2 = 1 / 32
3-2 = 1 / 9
Law 7 :
x0 = 1
Example :
30 = 1
Law 8 :
x1 = x
Example :
31 = 3
Law 9 :
xm/n = y -----> x = yn/m
Example :
x1/2 = 3
x = 32/1
x = 32
x = 9
Law 10 :
(x / y)-m = (y / x)m
Example :
(5 / 3)-2 = (3 / 5)2
(5 / 3)-2 = 32 / 52
(5 / 3)-2 = 9 / 25
Law 11 :
ax = by -----> x = y
Example :
3m = 35 -----> m = 3
Law 12 :
xa = ya -----> x = y
Example :
k3 = 53 -----> k = 5
Important Note
Many students do not know the difference between
(-3)2 and -32
Order of operations (PEMDAS) dictates that parentheses take precedence.
So, we have
(-3)2 = (-3) ⋅ (-3)
(-3)2 = 9
Without parentheses, exponents take precedence :
-32 = -3 ⋅ 3
-32 = -9
The negative is not applied until the exponent operation is carried through. We have to make sure that we understand this. So, we will not make this common mistake.
Sometimes, the result turns out to be the same, as in.
(-2)3 and -23
We have to make sure why they yield the same result.
Laws of Exponents - Examples
Example 1 :
If a-1/2 = 5, then find the value of a.
Solution :
a-1/2 = 5
a = 5-2/1
a = 5-2
a = 1/52
a = 1/25
Example 2 :
If n = 12 + 14 + 16 + 18 +...............+ 150, then find the value of n.
Solution :
1 to the power of anything is equal to 1.
So, we have
12 = 1
14 = 1
16 = 1
and so on.
List out all the exponents.
2, 4, 6, 8, ................ 48, 50
Divide each element by 2.
1, 2, 3, 4, ................ 24, 25
We can clearly see there are 25 terms in the series shown below.
12 + 14 + 16 + 18 +...............+ 150
Therefore, n is the sum of twenty-five 1's.
That is,
n = 12 + 14 + 16 + 18 +...............+ 150
n = 25
Example 3 :
If 42n + 3 = 8n + 5, then find the value of n.
Solution :
42n + 3 = 8n + 5
(22)2n + 3 = (23)n + 5
22(2n + 3) = 23(n + 5)
Equate the exponents.
2(2n + 3) = 3(n + 5)
4n + 6 = 3n + 15
n = 9
Example 4 :
If 2x / 2y = 23, then find the value x in terms of y.
Solution :
2x / 2y = 23
2x - y = 23
x - y = 3
x = y + 3
Example 5 :
If ax = b, by = c and cz = a, then find the value of xyz.
Solution :
Let
ax = b -----(1)
by = c -----(2)
cz = a -----(3)
Substitute a = cz in (1).
(1)-----> (cz)x = b
czx = b
Substitute c = by.
(by)zx = b
bxyz = b
bxyz = b1
xyz = 1
After having gone through the stuff given above, we hope that the students would have understood, "Laws of Exponents".
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