(a) \(\ 11^3 \)
(b) \(\ 2\times10^3 \)
(c) \(\ \left(\frac{1}{2}\right)^{-5} \)
(d) \(\ (-4)^{-2} \)
Solution:
(a) \(\ 11^3 \) = 11×11×11 = 1331
(b) \(\ 2\times10^3 \) = 2×10×10×10 = 2000
(c) \(\ \left(\frac{1}{2}\right)^{-5} = \frac{1^{-5}}{2^{-5}} = \frac{2^5}{1^5} = 2^5 = 32 \)
(d) \(\ (-4)^{-2} = \frac{1}{(-4)^2}=\frac{1}{16} \)
2. Express the following numbers in terms of powers of their prime factors.
(i) 729
From prime factorisation, we have
\(\ 729 = 3^6 \)
(ii) 3125
From prime factorisation, we have
\(\ 3125 = 5^5 \)
(iii) 3600
From prime factorisation, we have
\(\ 3600 = 2^4×3^2×5^5 \)
(iv) 108×192
From prime factorisation, we have
\(\ 108×192 = (2^2×3^3)×(2^6×3)=2^8×3^4 \)
3. Simplify-
(i) \(\ (-3)^2×(-5)^2 \)
Solution:
\(\ (-3)^2×(-5)^2 = 9×25=225 \)
(ii) \(\ (2^3×2)^4 \)
Solution:
\(\ (2^3×2)^4=2^{3×4}×2^4=2^{12}×2^4=2^{16} \)
(iii) \(\ 2^0×3^0×4^0 \)
Solution:
\(\ 2^0×3^0×4^0 = 1×1×1=1 \)
(iv) \(\ \left(\frac{5}{8}\right)^{-7}×\left(\frac{8}{5}\right)^{-4} \)
Solution:
\(\ \left(\frac{5}{8}\right)^{-7}×\left(\frac{8}{5}\right)^{-4} \)
\(\ = \left(\frac{8}{5}\right)^7×\left(\frac{8}{5}\right)^{-4} \)
\(\ = \left(\frac{8}{5}\right)^{7+(-4)} \)
\(\ = \left(\frac{8}{5}\right)^{7-4} \)
\(\ = \left(\frac{8}{5}\right)^3 \)
\(\ = \frac{512}{125} \)
4. Compare the following numbers.
(i) \(\ 2^8, 8^2 \)
Ans. \(\ 2^8 > 8^2 \)
(ii) \(\ 2.7×10^{12}, 1.5×10^8 \)
Ans. \(\ 2.7×10^{12} > 1.5×10^8 \)
5. Express the following with the help of positive power.
(i) \(\ (2)^{-3}×(-7)^{-3} \)
Solution:
\(\ (2)^{-3}×(-7)^{-3} = \frac{1}{2^3}×\frac{1}{(-7)^3}=\frac{1}{(-14)^3} \)
(ii) \(\ (2)^{-3}×(-7)^{-3} \)
Solution:
\(\ (-3)^{-4}×\left(\frac{5}{3}\right)^{-4} = \frac{1}{(-3)^4}×\left(\frac{3}{5}\right)^{4}=\frac{3^4}{(-3)^4×5^4} \)
\(= \left(\frac{3}{-15}\right)^{4} = \left(\frac{1}{-5}\right)^{4} = \frac{1}{5^4} \)
6. Express the following number in standard form.
(i) 3430000
Solution: \(\ 3.43×10^6 \)
(ii) 70040000000
Solution: \(\ 7.004×10^7 \)
(iii) 0.00000015
Solution: \(\ 1.5×10^{-7} \)
(iv) 0.00001436
Solution: \(\ 1.436×10^{-5} \)
7. Express the following in general form.
(i) \(\ 1.0001×10^9\)
Solution: 10001000000000
(ii) \(\ 3.02×10^{-6} \)
Solution: 0.000302
8. Find the value of \(\ m \) such that
\(\ (-3)^{m+1}×(-3)^5 = (-3)^7 \)
Solution:
\(\ (-3)^{m+1}×(-3)^5 = (-3)^7 \)
\(\ ⇨ (-3)^{m+1} = (-3)^7 ÷ (-3)^5 \)
\(\ ⇨ (-3)^{m+1} = (-3)^{7-5} \)
\(\ ⇨ (-3)^{m+1} = (-3)^2 \)
\(\ ⇨ m+1 = 2 \)
\(\ ⇨ m = 2-1 \)
\(\ ⇨ m = 1 \)
9. The size (diameter) of a plant cell is 0.00001275m. Express it in standard form.
Ans. \(\ 1.275×10^{-5} \)
10. In a shelf there are 5 books of thickness 20mm and 5 papers of thickness 0.016mm . what is the total thickness of the shelf?
Solution:
The thickness of 5 books = 5×20 mm = 100 mm
The thickness 5 papers = 5×0.016 mm = 0.080 mm
⸫ The total thickness of the shelf = (100+0.080) mm = 100.080 mm
11. Choose the correct options:
(a) The value of \(\ 3^{-3} \) is
(i) \(\ 3^3 \)
(ii) \(\ 3^{\frac{1}{3}} \)
(iii) \(\ \frac{1}{3^3} \)
(iv) \(\ 3×3 \)
Solution: (iii) \(\ \frac{1}{3^3} \)
(b) The value of \(\ \left(\frac{2}{3}\right)^2 \) is
(i) \(\ \frac{1}{(2×3)^2} \)
(ii) \(\ (2×3)^{-2} \)
(iii) \(\ \left(\frac{3}{2}\right)^{-2} \)
(iv) \(\ \left(\frac{3}{2}\right)^{\frac{1}{2}} \)
Solution: (iii) \(\ \left(\frac{3}{2}\right)^{-2} \)
(c) The value of \(\ \left(-\frac{2}{3}\right)^4 \) is
(i) \(\ \frac{8}{12} \)
(ii) \(\ \frac{16}{81} \)
(iii) \(\ -\frac{16}{81} \)
(iv) \(\ -\frac{8}{12} \)
Solution: (ii) \(\ \frac{16}{81} \)
(d) The standard form of 0.000064 is
(i) \(\ 64×10^4 \)
(ii) \(\ 64×10^{-4} \)
(iii) \(\ 6.4×10^5 \)
(iv) \(\ 6.4×10^{-5} \)
Solution: (iv) \(\ 6.4×10^{-5} \)
(e) The value of \(\ 2.03×10^{-5} \) is-
(i) 0.203
(ii) 0.0000203
(iii) 203000
(iv) 0.00203
Solution: (ii) 0.0000203
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