Exercise R4 | Class 10 | Mathematics | Revision Exercise

1. Find the value of-
(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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