There, we state the 7 Laws of indices and
the two Rules used for solving problems.
We also provide Links to the
explanations and proofs of the 7 laws.
Here, in Multiplying Exponents,
we apply the 7 Laws and the
2 Rules in solving problems.
Solved Example 1 : Multiplying Exponents
Find the value of
(i) (3⁄4)4 x (2⁄3)5 x (3⁄8)2
(ii) { (1⁄4)2 - (1⁄4)3} x 26
Solution:
(i)(3⁄4)4 x (2⁄3)5 x (3⁄8)2
(3⁄4)4 x (2⁄3)5 x (3⁄8)2
= (34⁄44) x (25⁄35) x (32⁄82)
{since (a⁄b)m = am⁄bm} (See Law 7 above)
= (34 x 4-4) x (25 x 3-5) x (32 x 8-2)
(since dividing with am is same as multiplying with a-m ) (See Law 3 above)
= (34 x 3-5 x 32) x (4-4 x 25 x 8-2)
(obtained by putting all factors of base 3 at one place and factors of base 4, 2, 8 at one place)
= (34 + (-5) + 2) x { (22)-4 x 25 x (23)-2}
(since am x a-n x ap = am + (-n) + p
and 4 = 22 and 8 = 23)
= ( 31) x { 22 x -4 x 25 x 23 x -2}
{since (am)-n = am x -n}
= 3 x { 2-8x 25 x 2-6}
= 3 x 2(-8) + 5 + (-6) = 3 x 2-9
= 3⁄29 = 3⁄512. Ans.
(ii) { (1⁄4)2 - (1⁄4)3} x 26
{ (1⁄4)2 - (1⁄4)3} x 26
= { (1⁄22)2 - (1⁄22)3} x 26
= { (2-2)2 - (2-2)3} x 26
= { 2-2 x 2 - 2-2 x 3} x 26
= { 2-4 - 2-6} x 26
= {2-4x 26- 2-6x 26}
= {2-4 + 6 - 2-6 + 6}
= {22 - 20}
= 4 - 1 = 3. Ans.
Solved Example 2 : Multiplying Exponents
By what number should we multiply 4-3
so that the product may be equal to 64?
Solution:
Let x be the number with which we should multiply 4-3
so that the product may be equal to 64.
Then x x 4-3 = 64.
⇒ x = 64⁄4-3
We know a⁄b-n = a x bn.
∴ x = 64 x 43
= 64 x (4 x 4 x 4) = 64 x 64 = 4096. Ans.
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By what number should (-2)-3 be multiplied
so that the product may be equal to 10-1?
Solution:
Let x be the number with which (-2)-3 be multiplied
so that the product may be equal to 10-1
Then x x (-2)-3 = 10-1.
⇒ x = 10-1⁄(-2)-3
We know a⁄b-n = a x bn.
∴ x = 10-1 x (-2)3
We know 10-1 = 1⁄10 and (-2)3 = -8
∴ x = (1⁄10) x -8 = -8⁄10 = -4⁄5. Ans.
By what number should (-25)-1 be divided
so that the quotient may be 5-1?
Solution:
Let x be the number with which (-25)-1 be divided
so that the quotient may be 5-1
Then (-25)-1⁄x = 5-1.
⇒ (-25)-1⁄5-1 = x.
⇒ {1⁄(-25)}⁄{1⁄(5)} = x.
⇒ 1⁄(-25) x 5 = x
⇒ x = -5⁄25 = -1⁄5. Ans.
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Solve the following problems on Multiplying Exponents :
Simplify (1⁄2)2 x (-2⁄3)3 x (-4⁄5)4
Find the value of (40 - 30) x 60
Fill in the blanks (i)54 x 57 = 5..... (ii) (5⁄2)6 x (2⁄5)6 = .........
Find the number to be multiplied by (-7)-1 so as to get (10)-1.
For Answers, see at the bottom of the page.
For more problems for practice on Application of Laws of exponents, go to Addition of Exponents.
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