Q1. Using laws of exponents, simplify and write the answer in exponential form
(i) 2^{3}×2^{4}×2^{5}
(ii) 5^{12}÷5^{3}
(iii) (7^{2})^{3}
(iv) (3^{2})^{5}÷34
(v) 3^{7}×2^{7}
(vi) (5^{21}÷5^{13})×5^{7}
Sol:
(i) 2^{3}×2^{4}×2^{5}
We know that, a^{m}+a^{n}+a^{p }= a^{m+n+p}
So, 2^{3}×2^{4}×2^{5} = 2^{3+4+5}
= 2^{12}
(ii) 5^{12}÷5^{3}
We know that, a^{m}÷a^{n }= a^{m−n}
So, 5^{12}÷5^{3} = 5^{12−3}
= 5^{9}
(iii) (7^{2})^{3}
We know that, (am)n=amn
So, (72)3 = 7(2)(3)
= 76
(iv) (3^{2})^{5}÷3^{4}
We know that, a^{m}÷a^{n }= a^{m−n} and (a^{m})^{n }= a^{mn}
So, (3^{2})^{5}÷3^{4} = 3^{10}÷3^{4}
= 3^{10−4}
= 3^{6}
(v) 3^{7}×2^{7}
We know that, (a^{m}×b^{m})=(a×b)^{m}
So, 3^{7}×2^{7} = (3×2)^{7}
= 6^{7}
(vi) (5^{21}÷5^{13})×5^{7}
We know that, a^{m}÷a^{n}=a^{m−n} and(a^{m}×a^{n})=(a)^{m+n}
So, (5^{21}÷5^{13})×5^{7} = (5^{21−13})×5^{7}
= (5^{8})×5^{7}
= 5^{8+7}
= 5^{15}
Q2. Simplify and express each of the following in exponential form
Sol:
Q3. Simplify and express each of the following in exponential form
Sol:
We know that,
We know that,
Q4. Write 9 ×9 ×9 ×9 ×9 in exponential form with base 3
Sol:
9 ×9 ×9 ×9 ×9 = (9)^{5} = (3^{2})^{5}
= 3^{10}
Q5. Simplify and write each of the following in exponential form
Sol:
Q6. Simplify
Sol:
Q7. Find the values of n in each of the following
Sol:
Equating the powers
= 2n + 3 = 11
= 2n = 11 3
= 2n = 8
= n = 4
(ii) 9×3^{n} = 3^{7}
= 3^{2}×3^{n} = 3^{7}
= 3^{2+n }= 3^{7}
Equating the powers
= 2 + n = 7
= n = 7 – 2
= n = 5
Equating the powers
= n + 5 = 5
= n = 0
Equating the powers
Equating the powers
= 4 + 5 = 2n + 1
= 2n + 1 = 9
= 2n = 8
= n = 4
= 3(2n – 2) =2(2n – 2)
= 6n – 6 = 4n – 4
= 6n – 4n = 6 – 4
= 2n = 2
= n = 1
Q8. find the value of n
Sol:
On equating the coefficient
3n – 15 = 3
3n = 3 + 15
3n = 12
n = 4
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