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.
a^{m }⋅ a^{n} = a^{m + n}
Example :
3^{4} ⋅ 3^{5} = 3^{4 + 5 }= 3^{9}
∙ Quotient Law
According to this law, when dividing two powers that have the same base we can subtract the exponents.
a^{m }÷ a^{n} = a^{m - n}
Example :
3^{9} ÷ 3^{2} = 3^{9 - 7 }= 3^{2}
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.
(a^{m })^{n} = a^{mn}
Example :
(3^{2})^{5} = 3^{2 x 5 }= 3^{10}
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} = p^{n} ⋅ q^{n}
Example :
(4 ⋅ 7)^{3} = 4^{3} ⋅ 7^{3}
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} = p^{n}/q^{n}
Example :
(2/3)^{4} = 2^{4}/3^{4}
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/y^{n}
Example :
5^{-2} = 1/5^{2}
5. For any nonzero base, if the exponent is zero, its value is 1.
x^{0} = 1
Example :
3^{0} = 1
6. For any base base, if there is no exponent, the exponent is assumed to be 1.
x^{1} = x
Example :
3^{1} = 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.
x^{m/n} = y ----> x = y^{n/m}
Example :
x^{2/3} = 3
x = 3^{3/2}
8. If two powers are equal with the same base, exponents can be equated.
a^{x} = a^{y} ----> x = y
Example :
3^{m} = 3^{5} ----> m = 5
9. If two powers are equal with the same exponent, bases can be equated.
x^{a} = y^{a} ----> x = y
Example :
k^{3} = 5^{3} ----> k = 5
(-4)^{2} and -4^{2 }are same ?
No, there is a difference between (-4)^{2} and -4^{2}.
In (-4)^{2}, order of operations (PEMDAS) says to take first.
(-4)^{2 }= (-4) ⋅ (-4)
(-4)^{2 }= 16
Without parentheses, exponents take precedence :
-4^{2 }= -4 ⋅ 4
-4^{2 }= -16
Sometimes, the result will be same as in (-2)^{3} and -2^{3}.
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) 3^{2}/3^{-2}.
Solution :
(i) 4^{-3 }:
= 1/4^{3}
= 1/(4 x 4 x 4)
= 1/64
(ii) 1/2^{-3 }:
= 2^{3}
= 2 x 2 x 2
= 8
(iii) (-2)^{5} x (-2)^{-3}^{ }:
= (-2)^{5 - 3}
= (-2)^{2}
= -2 x -2
= 4
(iv) 3^{2}/3^{-2}^{ }:
= 3^{2}/3^{-2}
= 3^{2} x 3^{2}
= 9 x 9
Example 2 :
Simplify and write the answer in exponential form:
(i) (3^{5 }÷ 3^{8})^{5} x 3^{-5} (ii) (-3)^{4 }x (5/3)^{4}
Solution :
(i) (3^{5 }÷ 3^{8})^{5} x 3^{-5 }:
= (3^{5 - }^{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}^{ }:
= 3^{4 }x 5^{4}/3^{4}
= 5^{4}
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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