Q.1. Simplify
(i)
(ii)
(iii)
(iv)
Ans.
(i)
=2^{1}
= 2
(ii)
(iii)
(iv)
= (6)^{3}
= 216
Q.2. Simplify:
(i)
(ii)
(iii)
Ans.
Q.3. Simplify:
(i) ^{31/4 }× 5^{1/4}
(ii) 2^{5/8} × 3^{5/8}
(iii) 6^{1/2} × 7^{1/2}
Ans.
Q.4. Simplify:
(i) (3^{4})^{1/4}
(ii) (3^{1/3})^{4}
(iii)
Ans.
Q.5. Evaluate
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Ans.
(i)
(ii)
(iii)
(iv)
(v)
(vi)
Q.6. If a = 2, b = 3, find the values of
(i) (a^{b} + b^{a})^{–1}
(ii) (a^{a} + b^{b})^{–1}
Ans.
(i) (a^{b} + b^{a})–1
(a^{b }+ b^{a}) ^{−1 }= (2^{3 }+ 3^{2}) ^{−1}
= (8 + 9)^{−1}
= (17)^{−1}
= 1/17
(ii) (a^{a} + b^{b})^{–1}
(a^{a }+ b^{b})^{−1 }= (2^{2 }+ 3^{3})^{−1}
= (4 + 27)^{−1}
= (31)^{−1}
=1/31
Q.7. Simplify
(i)
(ii) (14641)^{0.25}
(iii)
(iv)
Ans.
(ii) (14641)^{0.25}
(iii)
(iv)
Q.8. Evaluate
(i)
(ii)
(iii)
(iv)
Ans.
(i)
=4(6)^{2} + (4)^{3} + 2(3)
=144 + 64 + 6
=214
(ii)
(iii)
(iv)
Q.9. Evaluate
(i)
(ii)
(iii)
(iv)
Ans.
(i)
(ii)
(iii)
= 2
(iv)
Q.10. Prove that
(i)
(ii)
(iii)
Ans.
(i)
LHS =
= √2
= RHS
∴
(ii)
= RHS
∴
(iii)
Q.11. Simplify and express the result in the exponential form of x.
Ans.
Hence, the result in the exponential form is
Q.12. Simplify the product
Ans.
Q.13. Simplify
(i)
(ii)
(iii)
Ans.
(i)
(ii)
(iii)
Q.14. Find the value of x in each of the following.
(i)
(ii)
(iii)
(iv)5^{x−3 }× 3^{2x−8 }= 225
(v)
Ans.
(i)
Hence, the value of x is 6.
(ii)
Hence, the value of x is 22.
(iii)
Hence, the value of x is 5.
(iv) 5^{x−3}×3^{2x−8 }= 225
⇒5^{x−3} × 3^{2x−8} = (15)^{2}
⇒5^{x−3} × 3^{2x − 8} = 5^{2} × 3^{2}
⇒x − 3 = 2 and 2x − 8 = 2
⇒x = 2 + 3 and 2x = 2 + 8
⇒x = 5 and 2x = 10
⇒ x = 5 and x = 5
⇒ x = 5
Hence, the value of x is 5.
(v)
Hence, the value of x is
Q.15.
Prove that
(i)
(ii)
(iii)
(iv)
Ans.
(i)
Hence,
(ii)
=x^{0}
=1
= RHS
Hence,
(iii)
=x^{ab−ac−ba+bc}.x^{ac−bc}
=x^{−ac+bc}.x^{ac−bc}
=x^{−ac+bc+ac−bc}
=x^{0}
= 1
= RHS
Hence,
(iv)
= 1
= RHS
Q.16. If x is a positive real number and exponents are rational numbers, simplify
Ans.
Q.17. If = 1/27, prove that m – n = 1.
Ans.
Hence, m  n = 1
Q.18. Write the following in ascending order of magnitude.
Ans.
On Comparing (1), (2) and (3), we get
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