Points to Remember
• The numbers with negative exponents also obey the following laws:
(i) x^{m} * x^{n} = x^{m+n}
(ii) x^{m} ÷ x^{n} = x^{m}^{–n}
(iii) x^{m} * b^{m} = (xb)^{m}
(iv) x^{0} = 1
• A number is said to be in the standard form, if it is expressed as the product of a number between 1 and 10 and the integral power of 10.
• Very small numbers can be expressed in standard form using negative exponents.
WE KNOW THAT
When we write 5^{4}, it means 5 * 5 * 5 * 5, i.e. 5 is multiplied 4 times. So 5 is the base and 4 is the exponent. We read 5^{4} as “5 raised to the power 4’’.
We also know the following laws of exponents
(i) x^{m} ⋅ x^{n} = x^{m}^{+n}
(ii) x^{m} ÷ x^{n} = x^{m}^{–n}
(iii) (x^{m})^{n} = x^{m*n}
(iv) x^{m} * y^{m} = (xy)^{m}
The value of any number raised to 0 is 1, i.e. a^{0} = 1.
We express very small or very large numbers in standard form (i.e scientific notation) for
Example:
(ii) 3600000000000 = 36 * 10^{11} = 3.6 * 10^{12}
So, 3.6 * 10^{12} is the standard form.
POWER WITH NEGATIVE EXPONENTS
For a nonzero integer x, we have
or x^{–m} * x^{m} = 1
So x^{–m} is the reciprocal (or the multiplicative inverse) of x^{m} and vice versa.
For example: (i) Reciprocal of 8^{–7} = 8^{7} and
(ii) Reciprocal of 8^{7 }= 8^{–7}
Solved Examples
Ques 1: Find the multiplicative inverse of the following.
(i) 2^{–4}
(ii) 10^{–5}
(iii) 7^{–2}
(iv) 5^{–3 }
(v) 10^{–100}
Solution:
(i) The multiplicative inverse of 2^{–4} is 2^{4}.
(ii) The multiplicative inverse of 10^{–5} is 10^{5}
(iii) The multiplicative inverse of 7^{–2} is 7^{2}.
(iv) The multiplicative inverse of 5^{–3 }is 5^{3}.
(v) The multiplicative inverse of 10^{–100} is 10^{100}.
Ques 2: Find the value of
Solution: By using rule 2 and 3 –
Ques 3: Solve the following: (3)^{2} × (5/3)^{3}
Solution: (3)^{2} × (5/3)^{3}
= (3 × 3) × ( ( 5 × 5 × 5 ) / ( 3 × 3 × 3 ) )
= 9 × (125/27)
= (125/3)
Ques 4: If x11 = y0 and x=2y, then y is equal to
a. 1/2
b. 1
c. 1
d. 2
Solution: Option A. x^{11} = y^{0} => x^{11} = 1 => x = 1. Given, x = 2y hence, y = x/2 =1/2
Ques 5: By what number (4)^{3} be multiplied so that the product become 1/16?
Solution: 4 Simplest way to to solve this would be:
1/16 = 1/4^{2} = (4)^{2 }
(4)^{3} × 4 = (4)^{2}
Ques 6: What is the value of 6^{3} ?
a. 18
b. 216
c. 729
d. 1296
Solution: Option b.
6^{3}
= 6 × 6 × 6
= 36 × 6
= 216
Ques 7: What is the value of (2)^{5}?
a. 0.03125
b. 0.03125
c. 10
d. 32
Solution: Option A.
(2)^{5}
= 0.03125
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