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Laws of Exponents
1. Multiplying powers with same baseFor example: x^{2} × x^{3}, 2^{3} × 2^{5}, (3)^{2} × (3)^{4}In multiplication of exponents if the bases are same then we need to add the exponents. Consider the following: 1. 2^{3} × 2^{2}= (2 × 2 × 2) × (2 × 2) = 2^{3 + 2} = 2^{5} 2. 3^{4} × 3^{2} = (3 × 3 × 3 × 3) × (3 × 3) = 3^{4 + 2} = 3^{6} 3. (3)^{3} × (3)^{4} = [(3) × (3) × (3)] × [(3) × (3) × (3) × (3)] = (3)^{3 + 4} = (3)^{7} 4. m^{5} × m^{3} = (m × m × m × m × m) × (m × m × m) = m^{5 + 3} = m^{8} From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added. a^{m} × a^{n} = 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 a^{m} × a^{n} = 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 m^{5} × n^{7}, 2^{3} × 3^{4} For example: 1. 5^{3} ×5^{6} = (5 × 5 × 5) × (5 × 5 × 5 × 5 × 5 × 5) = 5^{3+6}[here the exponents are added] = 5^{9} 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. 3^{2} × 3^{5} = 3^{2 + 5} = 3^{7} 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. 2. Dividing powers with the same baseFor example:3^{5} ÷ 3^{1}, 2^{2} ÷ 2^{1}, 5(^{2}) ÷ 5^{3} In division if the bases are same then we need to subtract the exponents. Consider the following: 2^{7} ÷ 2^{4} = 2^{7}/2^{4} = (2 × 2 × 2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2) = 2^{7  4} = 2^{3} 5^{6} ÷ 5^{2} = 5^{6}/5^{2} = (5 × 5 × 5 × 5 × 5 × 5)/(5 × 5) = 5^{62} = 5^{4} 10^{5} ÷ 10^{3} = 10^{5}/10^{3} = (10 × 10 × 10 × 10 × 10)/(10 × 10 × 10) = 10^{5  3} = 10^{2} 7^{4} ÷ 7^{5} = 7^{4}/7^{5} = (7 × 7 × 7 × 7)/(7 × 7 × 7 × 7 × 7) = 7^{4  5} = 7^{1} Let a be a non zero number, then a^{5} ÷ a^{3} = a^{5}/a^{3} = (a × a × a × a × a)/(a × a × a) = a^{5  3} = a^{2} again, a^{3} ÷ a^{5} = a^{3}/a^{5} = (a × a × a)/(a × a × a × a × a) = a^{(5  3 )} = a^{2} Thus, in general, for any nonzero integer a, a^{m} ÷ a^{n} = a^{m}/a^{n} = a^{m  n} Note 1: Where m and n are whole numbers and m > n; a^{m} ÷ a^{n} = a^{m}/a^{n} = 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 a^{m} ÷ a^{n} = a^{m  n} if m < n, then a^{m} ÷ a^{n} = 1/a^{n  m} Similarly, (a/b)^{m} ÷ (a/b)^{n} = (a/b)^{m n} For example: 3. Power of a powerFor example: (2^{3})^{2}, (5^{2})^{6}, (3^{2} )^{3}In power of a power you need multiply the powers. Consider the following (i) (2^{3})^{4} Now, (2^{3})^{4} means 2^{3} is multiplied four times i.e. (2^{3})^{4} = 2^{3} × 2^{3} × 2^{3} × 2^{3} =2^{3 + 3 + 3 + 3} =2^{12} Note: by law (l), since a^{m} × a^{n} = a^{m + n}. (ii) (2^{3})^{2} Similarly, now (2^{3})^{2} means 2^{3} is multiplied two times i.e. (2^{3})^{2} = 2^{3} × 2^{3} = 2^{3 + 3} [since a^{m} × a^{n} = a^{m + n}] = 2^{6} Note: Here, we see that 6 is the product of 3 and 2 i.e,(2^{3})^{2} = 2^{3 × 2} = 2^{6} (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.(3^{2})^{4} = 3^{2 × 4} = 3^{8} 2. (5^{3})^{6} = 5^{3 × 6} = 5^{18} 3. (4^{3})^{8} = 4^{3 × 8} = 4^{24} 4. (a^{m})^{4} = a^{m × 4} = a^{4m} 5. (2^{3})^{6} = 2^{3 × 6} = 2^{18} 6. (x^{m})^{n} = x^{m × (n)} = x^{mn} 7. (5^{2})^{7} = 5^{2 × 7} = 5^{14} 8. [(3)^{4}]^{2} = (3)^{4 × 2} = (3)^{8} In general, for any noninteger a, (a^{m})^{n}=a^{m × n} =a^{mn}.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} 4. Multiplying powers with the same exponentsFor example: 3^{2} × 2^{2}, 5^{3} × 7^{3}We consider the product of 4^{2} and 3^{2}, which have different bases, but the same exponents. (i) 4^{2} × 3^{2} [here the powers are same and the bases are different] = (4 × 4)×(3 × 3) = (4 × 3)×(4 × 3) = 12 × 12 = 12^{2} Here, we observe that in 12^{2}, the base is the product of bases 4 and 3. We consider, (ii) 4^{3} × 2^{3} =(4 × 4 × 4)×(2 × 2 × 2) =(4 × 2)× ( 4 × 2)× (4 × 2) =8 × 8 × 8 =8^{3} (iii) We also have, 2^{3} × a^{3} = (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, 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] Note: In general, for any nonzero integer a, b. a^{m} × b^{m} = (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, () × () = +] 5. Negative ExponentsIf 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, thena^{m} is the reciprocal of a^{m}, i.e., a^{m} = 1/a^{m}, if we take ‘a’ as p/q then (p/q)^{m} = 1/(p/q)^{m} = (q/p)^{m} again, 1/a^{m} = a^{m} Similarly, (a/b)^{n} = (b/a)^{n}, where n is a positive integer Consider the following 2^{1} = 1/2 2^{2} = 1/2^{2} = 1/2 × 1/2 = 1/4 2^{3} = 1/2^{3} = 1/2 × 1/2 × 1/2 = 1/8 2^{4} = 1/2^{4} = 1/2 × 1/2 × 1/2 × 1/2 = 1/16 2^{5} = 1/2^{5} = 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} = 1/10^{3} [here we can see that 1 is in the numerator and in the denominator 10^{3} 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} = 1/2^{5} = 1/2 × 1/2 = 1/4 4. 1/3^{4} = 3^{4} = 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} 6. Power with exponent zeroIf 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 a^{m} × a^{n} = a^{m+ n}] = (2/3)^{3  3} = (2/3)^{0} = 1 2. 2^{5} ÷ 2^{5} = 2^{5}/2^{5} = (2 × 2 × 2 × 2 × 2)/(2 × 2 × 2 × 2 × 2) = 2^{5  5} [here by the law a^{m} ÷ a^{n} =a^{m  n}] = 2^{0} = 1 3. 4^{0} × 3^{0} = 1 × 1[here as we know anything to the power 0 is 1] = 1 4. a^{m} × 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 7. Fractional ExponentIn fractional exponent we observe that the exponent is in fraction form.Consider the following: 2^{1/1} = 2 (it will remain 2). 2^{1/2} =√2 (square root of 2). 2^{1/3} =∛2 (cube root of 2). 2^{1/4} =∜2 (fourth root of 2). 2^{1/5} =^{5}√2 (fifth root of 2). For example: Exponents  Worksheets


