Laws of exponents arise out of certain basic ideas. A positive exponent of a number indicate how many times we use that number in a multiplication whereas a negative exponent suggests us how many times we use that number in a division, since the opposite of multiplying is dividing.
∙ Product law
According to this law, when multiplying two powers that have the same base, we can add the exponents.
am ⋅ an = am + n
Example :
34 ⋅ 35 = 34 + 5 = 39
∙ Quotient Law
According to this law, when dividing two powers that have the same base we can subtract the exponents.
am ÷ an = am - n
Example :
39 ÷ 32 = 39 - 7 = 32
where a (a ≠ 0), m, n are integers. Note that the base should be the same in both the quantities. How does it work? Study the following examples.
∙ Power Law
According to this law, when raising a power to another power, we can just multiply the exponents.
(am )n = amn
Example :
(32)5 = 32 x 5 = 310
Some other laws of exponents :
1. A product two or more numbers raised to a power equals the product of each number raised to that power.
If p and q are any nonzero real numbers and n is any integer, then
(pq)n = pn ⋅ qn
Example :
(4 ⋅ 7)3 = 43 ⋅ 73
2. A quotient raised to a positive power equals the quotient of each base raised to that power.
If p and q are any nonzero real numbers and n is a positive integer, then
(p/q)n = pn/qn
Example :
(2/3)4 = 24/34
3. A quotient raised to a negative power equals the reciprocal of the quotient raised to the opposite (positive) power.
If p and q are any nonzero real numbers and m is a positive integer, then
(p/q)-m = (q/p)m
Example :
(5/3)-2 = (3/5)2
4. If a power is moved from numerator to denominator or denominator to numerator, the sign of the exponent has to be changed.
y-n = 1/yn
Example :
5-2 = 1/52
5. For any nonzero base, if the exponent is zero, its value is 1.
x0 = 1
Example :
30 = 1
6. For any base base, if there is no exponent, the exponent is assumed to be 1.
x1 = x
Example :
31 = 3
7. If an exponent is transferred from one side of the equation to the other side of the equation, reciprocal of the exponent has to be taken.
xm/n = y ----> x = yn/m
Example :
x2/3 = 3
x = 33/2
8. If two powers are equal with the same base, exponents can be equated.
ax = ay ----> x = y
Example :
3m = 35 ----> m = 5
9. If two powers are equal with the same exponent, bases can be equated.
xa = ya ----> x = y
Example :
k3 = 53 ----> k = 5
(-4)2 and -42 are same ?
No, there is a difference between (-4)2 and -42.
In (-4)2, order of operations (PEMDAS) says to take first.
(-4)2 = (-4) ⋅ (-4)
(-4)2 = 16
Without parentheses, exponents take precedence :
-42 = -4 ⋅ 4
-42 = -16
Sometimes, the result will be same as in (-2)3 and -23.
For a negative number with odd exponent, the result is always negative.
Example 1 :
Find the value of (i) 4-3 (ii) 1/2-3 (iii) (-2)5 x (-2)-3 (iv) 32/3-2.
Solution :
(i) 4-3 :
= 1/43
= 1/(4 x 4 x 4)
= 1/64
(ii) 1/2-3 :
= 23
= 2 x 2 x 2
= 8
(iii) (-2)5 x (-2)-3 :
= (-2)5 - 3
= (-2)2
= -2 x -2
= 4
(iv) 32/3-2 :
= 32/3-2
= 32 x 32
= 9 x 9
Example 2 :
Simplify and write the answer in exponential form:
(i) (35 ÷ 38)5 x 3-5 (ii) (-3)4 x (5/3)4
Solution :
(i) (35 ÷ 38)5 x 3-5 :
= (35 - 8)5 x 3-5
= (3-3)5 x 3-5
= 3-3 x 5 x 3-5
= 3-15 x 3-5
= 3-15 - 5
= = 3-20
(ii) (-3)4 x (5/3)4 :
= 34 x 54/34
= 54
Example 3 :
Find x so that (-7)x + 2 x (-7)5 = (-7)10.
Solution :
(-7)x + 2 x (-7)5 = (-7)10
(-7)x + 2 + 5 = (-7)10
(-7)x + 7 = (-7)10
Since the bases are equal, we can equate the exponents.
x + 7 = 10
Subtract 7 from each side.
x = 3
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