Question 1

Q1: Evaluate. (i) $3^{-2}$ (ii) $(-4)^{-2}$ (iii) $(\frac{1}{2})^{-5}$

Accepted Answer

(i) $3^{-2}$ Formula: $a^{-m} = \frac{1}{a^m}$ Solution: $$3^{-2} = \frac{1}{3^2} = \frac{1}{3 	imes 3} = \frac{1}{9}$$ Final Answer: $\frac{1}{9}$ ---  (ii) $(-4)^{-2}$ Formula: $a^{-m} = \frac{1}{a^m}$ Solution: $$(-4)^{-2} = \frac{1}{(-4)^2} = \frac{1}{(-4) 	imes (-4)} = \frac{1}{16}$$ Final An...

Question 2

Q2: Simplify and express the result in power notation with positive exponent. (i) $(-4)^{5} \div (-4)^{8}$ (ii) $(\frac{1}{2^{3}})^{2}$ (iii) $(-3)^{4} 	...

Accepted Answer

(i) $(-4)^{5} \div (-4)^{8}$ Formula: $a^m \div a^n = a^{m-n}$ and $a^{-m} = \frac{1}{a^m}$ Solution: $$(-4)^{5} \div (-4)^{8} = (-4)^{5-8} = (-4)^{-3} = \frac{1}{(-4)^3}$$ Final Answer: $\frac{1}{(-4)^3}$ ---  (ii) $(\frac{1}{2^{3}})^{2}$ Formula: $(a^m)^n = a^{mn}$ Solution: $$(\frac{1}{2^3})^2 =...

Question 3

Q3: Find the value of. (i) $(3^{\circ}+4^{-1}) 	imes 2^{2}$ (ii) $(2^{-1} 	imes 4^{-1}) \div 2^{-2}$ (iii) $(\frac{1}{2})^{-2}+(\frac{1}{3})^{-2}+(\frac...

Accepted Answer

(i) $(3^{\circ}+4^{-1}) 	imes 2^{2}$ Formulas: $a^0 = 1$ and $a^{-m} = \frac{1}{a^m}$ Solution: $$(3^0 + 4^{-1}) 	imes 2^2 = (1 + \frac{1}{4}) 	imes 4 = (\frac{4+1}{4}) 	imes 4 = \frac{5}{4} 	imes 4 = 5$$ Final Answer: 5 ---  (ii) $(2^{-1} 	imes 4^{-1}) \div 2^{-2}$ Solution: $$(2^{-1} 	imes...

Question 4

Q4: Evaluate (i) $\frac{8^{-1} 	imes 5^{3}}{2^{-4}}$ (ii) $(5^{-1} 	imes 2^{-1}) 	imes 6^{-1}$

Accepted Answer

(i) $\frac{8^{-1} 	imes 5^{3}}{2^{-4}}$ Solution: First, express 8 as a power of 2: $8 = 2^3$. $$\frac{8^{-1} 	imes 5^3}{2^{-4}} = \frac{(2^3)^{-1} 	imes 5^3}{2^{-4}} = \frac{2^{-3} 	imes 5^3}{2^{-4}}$$ Using the law $\frac{a^m}{a^n} = a^{m-n}$: $$= 2^{-3 - (-4)} 	imes 5^3 = 2^{-3+4} 	imes 5^3...

Question 5

Q5: Find the value of $m$ for which $5^{m} \div 5^{-3}=5^{5}$.

Accepted Answer

Given: $5^m \div 5^{-3} = 5^5$ To Find: The value of $m$. Formula: $a^m \div a^n = a^{m-n}$ Solution: $$5^m \div 5^{-3} = 5^5$$ $$5^{m - (-3)} = 5^5$$ $$5^{m+3} = 5^5$$ On both sides, the powers have the same base (5), so their exponents must be equal. $$m+3 = 5$$ $$m = 5 - 3$$ $$m = 2$$ Final Answe...

Question 6

Q6: Evaluate (i) $\{(\frac{1}{3})^{-1}-(\frac{1}{4})^{-1}\}^{-1}$ (ii) $(\frac{5}{8})^{-7} 	imes (\frac{8}{5})^{-4}$

Accepted Answer

(i) $\{(\frac{1}{3})^{-1}-(\frac{1}{4})^{-1}\}^{-1}$ Formula: $(\frac{a}{b})^{-m} = (\frac{b}{a})^m$ Solution: $$\{(\frac{1}{3})^{-1} - (\frac{1}{4})^{-1}\}^{-1} = \{(\frac{3}{1})^1 - (\frac{4}{1})^1\}^{-1} = \{3 - 4\}^{-1} = (-1)^{-1} = \frac{1}{-1} = -1$$ Final Answer: -1 ---  (ii) $(\frac{5}{8})^...

Question 7

Q7: Simplify. (i) $\frac{25 	imes t^{-4}}{5^{-3} 	imes 10 	imes t^{-8}}(t 
eq 0)$ (ii) $\frac{3^{-5} 	imes 10^{-5} 	imes 125}{5^{-7} 	imes 6^{-5}}$

Accepted Answer

(i) $\frac{25 	imes t^{-4}}{5^{-3} 	imes 10 	imes t^{-8}}(t 
eq 0)$ Solution: First, express all numbers as powers of prime factors: $25 = 5^2$ $10 = 2 	imes 5$ The expression becomes: $$\frac{5^2 	imes t^{-4}}{5^{-3} 	imes (2 	imes 5) 	imes t^{-8}} = \frac{5^2 	imes t^{-4}}{5^{-3} 	imes...

Question 8

Q1: Express the following numbers in standard form. (i) 0.0000000000085 (ii) 0.00000000000942 (iii) 6020000000000000 (iv) 0.00000000837 (v) 31860000000

Accepted Answer

(i) 0.0000000000085 Solution: To write the number in standard form ($k 	imes 10^n$, where $1 \le k < 10$), we move the decimal point to the right until there is one non-zero digit to its left. Here, we move the decimal point 12 places to the right. $$0.0000000000085 = 8.5 	imes 10^{-12}$$ Final An...

Question 9

Q2: Express the following numbers in usual form. (i) $3.02 	imes 10^{-6}$ (ii) $4.5 	imes 10^{4}$ (iii) $3 	imes 10^{-8}$ (iv) $1.0001 	imes 10^{9}$ (...

Accepted Answer

(i) $3.02 	imes 10^{-6}$ Solution: The exponent is -6, so we move the decimal point 6 places to the left. $$3.02 	imes 10^{-6} = 0.00000302$$ Final Answer: 0.00000302 ---  (ii) $4.5 	imes 10^{4}$ Solution: The exponent is 4, so we move the decimal point 4 places to the right. $$4.5 	imes 10^4 =...

Question 10

Q3: Express the number appearing in the following statements in standard form. (i) 1 micron is equal to $\frac{1}{1000000} \mathrm{~m}$. (ii) Charge of an...

Accepted Answer

(i) 1 micron is equal to $\frac{1}{1000000} \mathrm{~m}$ Solution: $$1 	ext{ micron} = \frac{1}{1000000} 	ext{ m} = \frac{1}{10^6} 	ext{ m} = 1 	imes 10^{-6} 	ext{ m}$$ Final Answer: $1 	imes 10^{-6} 	ext{ m}$ ---  (ii) Charge of an electron is $0.000,000,000,000,000,000,16$ coulomb. Solution...

Question 11

Q4: In a stack there are 5 books each of thickness 20 mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack.

Accepted Answer

Given: Number of books = 5 Thickness of one book = 20 mm Number of paper sheets = 5 Thickness of one paper sheet = 0.016 mm To Find: The total thickness of the stack. Solution: First, calculate the total thickness of the books: $$	ext{Thickness of 5 books} = 5 	imes 20 	ext{ mm} = 100 	ext{ mm}$...

NCERT Solutions