The laws of exponents are explained here along with their examples.
For example: x² × x³, 2³ × 2⁵, (3)² × (3)⁴
In multiplication of exponents if the bases are same then we need to add the exponents.
Consider the following:
1. 2³ × 2²= (2 × 2 × 2) × (2 × 2) = 2\(^{3 + 2}\) = 2⁵
2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 3\(^{4 + 2}\) = 3⁶
3. (3)³ × (3)⁴ = [(3) × (3) × (3)] × [(3) × (3) × (3) × (3)] = (3)\(^{3 + 4}\) = (3)⁷
4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m) = m\(^{5 + 3}\) = m⁸
From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.
aᵐ × aⁿ = a\(^{m + n}\)
In other words, if ‘a’ is a nonzero integer or a nonzero rational number and m and n are positive integers, then
aᵐ × aⁿ = a\(^{m + n}\)
Similarly, (a/b)ᵐ × (a/b)ⁿ = (a/b)\(^{m + n}\)
Note:
(i) Exponents can be added only when the bases are same.
(ii) Exponents cannot be added if the bases are not same like
m⁵ × n⁷, 2³ × 3⁴
For example:
1. 5³ ×5⁶
= (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5 × 5)
= 5\(^{3 + 6}\), [here the exponents are added]
= 5⁹
2. (7)\(^{10}\) × (7)¹²
= [(7) × (7) × (7) × (7) × (7) × (7) × (7) × (7) × (7) × (7)] × [( 7) × (7) × (7) × (7) × (7) × (7) × (7) × (7) × (7) × (7) × (7) × (7)].
= (7)\(^{10 + 12}\), [exponents are added]
= (7)²²
3. (1/2)⁴ × ( 1/2)³
=[(1/2) × ( 1/2) × ( 1/2) × ( 1/2)] × [ ( 1/2) × ( 1/2) × ( 1/2)]
=(1/2)\(^{4 + 3}\)
=(1/2)⁷
4. 3² × 3⁵
= 3\(^{2 + 5}\)
= 3⁷
5. (2)⁷ × (2)³
= (2)\(^{7 + 3}\)
= (2)\(^{10}\)
6. (4/9)³ × (4/9)²
= (4/9)\(^{3 + 2}\)
= (4/9)⁵
We observe that the two numbers with the same base are
multiplied; the product is obtained by adding the exponent.
For example:
3⁵ ÷ 3¹, 2² ÷ 2¹, 5(²) ÷ 5³
In division if the bases are same then we need to subtract the exponents.
Consider the following:
2⁷ ÷ 2⁴ = 2⁷/2⁴ = (2 × 2 × 2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2) = 2\(^{7  4}\) = 2³
5⁶ ÷ 5² = 5⁶/5² = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5) = 5\(^{6  2}\) = 5⁴
10⁵ ÷ 10³ = 10⁵/10³ = (10 × 10 × 10 × 10 × 10)/(10 × 10 × 10) = 10\(^{5  3}\) = 10²
7⁴ ÷ 7⁵ = 7⁴/7⁵ = (7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7) = 7\(^{4  5}\) = 7\(^{1}\)
Let a be a non zero number, then
a⁵ ÷ a³ = a⁵/a³ = (a × a × a × a × a)/(a × a × a) = a\(^{5  3}\) = a²
again, a³ ÷ a⁵ = a³/a⁵ = (a × a × a)/(a × a × a × a × a) = a\(^{(5  3)}\)
= a\(^{2}\)
Thus, in general, for any nonzero integer a,
aᵐ ÷ aⁿ = aᵐ/aⁿ = a\(^{m  n}\)
Note 1:
Where m and n are whole numbers and m > n;
aᵐ ÷ aⁿ = aᵐ/aⁿ = a\(^{(n  m)}\)
Note 2:
Where m and n are whole numbers and m < n;
We can generalize that if ‘a’ is a nonzero integer or a nonzero rational number and m and n are positive integers, such that m > n, then
aᵐ ÷ aⁿ = a\(^{m  n}\) if m < n, then aᵐ ÷ aⁿ = \(\frac{1}{a^{n  m}}\)
Similarly, (a/b)ᵐ ÷ (a/b)ⁿ = (a/b)\(^{m  n}\)
For example:
1. 7\(^{10}\) ÷ 7⁸ = \(\frac{7^{10}}{7^{8}}\)
= (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)
= 7\(^{10  8}\), [here exponents are subtracted]
= 7²
2. p⁶ ÷ p¹=p⁶/p¹
= (p × p × p × p × p × p)/p
= p\(^{6  1}\), [here exponents are subtracted]
= p⁵
3. 4⁴ ÷ 4² = 4⁴/4²
= (4 × 4 × 4 × 4)/(4 × 4)
= 4\(^{4  2}\), [here exponents are subtracted]
= 4²
4. 10² ÷ 10⁴ = 10²/10⁴
= (10 × 10)/(10 × 10 × 10 × 10)
= 10\(^{(4  2)}\), [See note (2)]
= 10\(^{2}\)
5. 5³ ÷ 5¹
= 5\(^{3  1}\)
= 5²
6. 3⁵/3²
= 3\(^{5  2}\)
= 3³
7. (5)⁹/(5)⁶
= (5)\(^{9  6}\)
= (5)³
8. (7/2)⁸ ÷ (7/2)⁵
= (7/2)\(^{8  5}\)
= (7/2)³
For example: (2³)², (5²)⁶, (3² )\(^{3}\)
In power of a power you need multiply the powers.
Consider the following
(i) (2³)⁴
Now, (2³)⁴ means 2³ is multiplied four times
i.e. (2³)⁴ = 2³ × 2³ × 2³ × 2³
=2\(^{3 + 3 + 3 + 3}\)
=2¹²
Note: by law (l), since aᵐ × aⁿ = a\(^{m + n}\).
(ii) (2³)²
Similarly, now (2³)² means 2³ is multiplied two times
i.e. (2³)² = 2³ × 2³
= 2\(^{3 + 3}\), [since aᵐ × aⁿ = a\(^{m + n}\)]
= 2⁶
Note: Here, we see that 6 is the product of 3 and 2 i.e,(2³)² = 2\(^{3 × 2}\)= 2⁶
(iii) (4\(^{ 2}\))³
Similarly, now (4\(^{2}\))³ means 4\(^{2}\)
is multiplied three times
i.e. (4\(^{2}\))³ =4\(^{2}\) × 4\(^{2}\) × 4\(^{2}\)
= 4\(^{2 + (2) + (2)}\)
= 4\(^{2  2  2}\)
= 4\(^{6}\)
Note: Here, we see that 6 is the product of 2 and 3 i.e, (4\(^{2}\))³ = 4\(^{2 × 3}\) =4\(^{6}\)
For example:
1.(3²)⁴ = 3\(^{2 × 4}\) = 3⁸
2. (5³)⁶ = 5\(^{3 × 6}\) = 5¹⁸
3. (4³)⁸ = 4\(^{3 × 8}\) = 4²⁴
4. (aᵐ)⁴ = a\(^{m × 4}\) = a⁴ᵐ
5. (2³)⁶ = 2\(^{3 × 6}\) = 2¹⁸
6. (xᵐ)\(^{n}\) = x\(^{m × (n)}\) = x\(^{mn}\)
7. (5²)⁷ = 5\(^{2 × 7}\) = 5¹⁴
8. [(3)⁴]² = (3)\(^{4 × 2}\) = (3)⁸
In general, for any noninteger a, (aᵐ)ⁿ=a\(^{m × n}\) = aᵐⁿ. Thus where m and n are whole numbers.
If ‘a’ is a nonzero rational number and m and n are positive integers, then {(a/b)ᵐ}ⁿ = (a/b) ᵐⁿ
For example:
[(2/5)³]²
= (2/5)\(^{3 × 2}\)
= (2/5)⁶
For example: 3² × 2², 5³ × 7³
We consider the product of 4² and 3², which have different bases, but the same exponents.
(i) 4² × 3² [here the powers are same and the bases are different]
= (4 × 4)×(3 × 3)
= (4 × 3)×(4 × 3)
= 12 × 12
= 12²
Here, we observe that in 12², the base is the product of bases 4 and 3.
We consider,
(ii) 4³ × 2³
=(4 × 4 × 4)×(2 × 2 × 2)
=(4 × 2)× ( 4 × 2)× (4 × 2)
=8 × 8 × 8
=8³
(iii) We also have, 2³ × a³
= (2× 2 × 2)×(a × a × a)
= (2 × a)×(2 × a)×(2 × a)
= (2 × a)³
= (2a)³ [here 2 × a = 2a]
(iv) Similarly, we have, a³ × b³
= (a × a × a)×(b × b × b)
= (a × b)× (a × b)× (a × b)
= (a × b)³
= (ab)³ [here a × b = ab]
Note: In general, for any nonzero integer a, b.
aᵐ × bᵐ
= (a × b)ᵐ
= (ab)ᵐ [here a × b = ab]
Note: Where m is any whole number.
(a)³ × (b)³
= [(a) × (a) × (a)]×[(b) × (b) × (b)]
= [(a) × (b)]× [(a) × (b)]× [(a) × (b)]
= [(a)×(b)]³
= (ab)³ [here a × b = ab and two negative become positive,
() × () = +]
If the exponent is negative we need to change it into positive exponent by writing the same in the denominator and 1 in the numerator.
If ‘a’ is a nonzero integer or a nonzero rational number and m is a positive integers, then
a\(^{m}\) is the reciprocal of aᵐ, i.e.,
a\(^{m}\) = \(\frac{1}{a^{m}}\), if we take ‘a’ as p/q then (p/q)\(^{m}\) = 1/(p/q)ᵐ = (q/p)ᵐ
again, \(\frac{1}{a^{m}}\) = aᵐ
Similarly, (a/b)\(^{n}\) = (b/a)ⁿ, where n is a positive integer
Consider the following
2\(^{1}\) = 1/2
2\(^{2}\) = 1/2² = 1/2 × 1/2 = 1/4
2\(^{3}\) = 1/2³ = 1/2 × 1/2 × 1/2 = 1/8
2\(^{4}\) = 1/2⁴ = 1/2 × 1/2 × 1/2 × 1/2 = 1/16
2\(^{5}\) = 1/2⁵ = 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = 1/32
[So in negative exponent we need to write 1 in the numerator and in the denominator 2 multiplied to itself five times as 2\(^{5}\). In other words negative exponent is the reciprocal of positive exponent]
For example:
1. 10\(^{3}\)
= \(\frac{1}{10^{3}}\), [here we can see that 1 is in the numerator and in the denominator 10³ as we know that negative exponent is the reciprocal]
= 1/10 × 1/10 × 1/10 [here 10 is multiplied to itself 3 times]
= 1/1000
2. (2)\(^{4}\)
= 1/(2)⁴ [here we can see that 1 is in the numerator and in the denominator (2)⁴]
=( 1/2) × ( 1/2) × ( 1/2) × ( 1/2)
= 1/16
3. 2\(^{5}\)
= 1/2⁵
= 1/2 × 1/2
= 1/4
4. \(\frac{1}{3^{4}}\)
= 3⁴
= 3 × 3 × 3 × 3
= 81
5. (7)\(^{3}\)
= 1/(7)³
6. (3/5)\(^{3}\)
= (5/3)³
7. (7/2)\(^{2}\)
= (2/7)²
If the exponent is 0 then you get the result 1 whatever the base is.
For example: 8\(^{0}\), ( a/b)\(^{0}\), m\(^{0}\)…....
If ‘a’ is a nonzero integer or a nonzero rational number then,
a\(^{0}\) = 1
Similarly, (a/b)\(^{0}\) = 1
Consider the following
a\(^{0}\) = 1 [anything to the power 0 is 1]
(a/b)\(^{0}\) = 1
(2/3)\(^{0}\) = 1
(3)\(^{0}\) = 1
For example:
1. (2/3)³ × (2/3)\(^{3}\)
= (2/3)\(^{3 + (3)}\), [here we know that aᵐ × aⁿ = a\(^{m + n}\)]
= (2/3)\(^{3  3}\)
= (2/3)\(^{0}\)
= 1
2. 2⁵ ÷ 2⁵
= 2⁵/2⁵
= (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2)
= 2\(^{5  5}\), [here by the law aᵐ ÷ aⁿ =a\(^{m  n}\)]
= 2
= 1
3. 4\(^{0}\) × 3\(^{0}\)
= 1 × 1, [here as we know anything to the power 0 is 1]
= 1
4. aᵐ × a\(^{m}\)
= a\(^{m  m}\)
= a\(^{0}\)
= 1
5. 5\(^{0}\) = 1
6. (4/9)\(^{0}\) = 1
7. (41)\(^{0}\) = 1
8. (3/7)\(^{0}\) = 1
In fractional exponent we observe that the exponent is in fraction form.
Consider the following:
2\(^{\frac{1}{1}}\) = 2 (it will remain 2).
2\(^{\frac{1}{2}}\) =√2 (square root of 2).
2\(^{\frac{1}{3}}\) =∛2 (cube root of 2).
2\(^{\frac{1}{4}}\) =∜2 (fourth root of 2).
2\(^{\frac{1}{5}}\) =⁵√2 (fifth root of 2).
For example:
1. a\(^{\frac{1}{n}}\), [Here a is called the base and 1/n is called the exponent or power]
= ⁿ√a [nth root of a]
2. 3\(^{\frac{1}{2}}\) = √3 [square root of 3]
3. 5\(^{\frac{1}{3}}\) = ∛5 [cube root of 5]
● Exponents
Integral Exponents of a Rational Numbers
● Exponents  Worksheets
8th Grade Math Practice
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