CBSE Class 8 Mathematics Power Play MCQs Set 11

Answer: (b) 9 is the base and 6 is the exponent

Explanation:
1. In a power, the big number written on the line is the base and the small raised number is the exponent.
2. So in 9⁶ the base is 9 and the exponent is 6.
3. 9⁶ means 9 multiplied by itself 6 times, which is not the same as 9 × 6.

Teacher's Note:
1. Base sits on the line; exponent sits up high.
2. The exponent counts how many times the base is used as a factor.

Answer: (d) 81

Explanation:
1. 3⁴ = 3 × 3 × 3 × 3.
2. 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81.
3. 12 is 3 × 4 and 27 is 3³, so those options are wrong.

Teacher's Note:
1. Never multiply the base by the exponent.
2. Multiply step by step and count the factors.

Answer: (b) 1

Explanation:
1. By the division law, n^a ÷ n^a = n^{a − a} = n⁰.
2. But any non-zero number divided by itself is 1.
3. So n⁰ = 1.

Teacher's Note:
1. The zero power of any non-zero number is 1, not 0.
2. This comes straight from dividing a power by itself.

Answer: (c) \( \frac{1}{8} \)

Explanation:
1. A negative exponent means the reciprocal: 2⁻³ = \( \frac{1}{2^{3}} \).
2. 2³ = 8, so 2⁻³ = \( \frac{1}{8} \).
3. The minus sign in the exponent does not make the number negative.

Teacher's Note:
1. Negative exponent means 'one over', not 'negative'.
2. Students often pick −8 or \( \frac{1}{6} \); check with the rule.

Answer: (c) \( n^{a+b} \)

Explanation:
1. n^a has a factors of n and n^b has b factors of n.
2. Multiplying puts all the factors together: a + b factors.
3. So \( n^{a} \times n^{b} = n^{a+b} \), which is option (c).

Teacher's Note:
1. Same base, multiply: add the exponents.
2. Multiplying exponents is for a power of a power.

Answer: (a) 128

Explanation:
1. The bases are the same, so add the exponents: 2⁴ × 2³ = 2⁴⁺³ = 2⁷.
2. 2⁷ = 128.
3. Multiplying the exponents (2¹² = 4096) is the common mistake.

Teacher's Note:
1. Check: 16 × 8 = 128.
2. Add exponents when multiplying powers of the same base.

Answer: (a) Only Statement I is true

Explanation:
1. (−1)⁷ has 7 negative factors; an odd number of negative factors gives a negative product, so (−1)⁷ = −1. Statement I is true.
2. (−2)⁴ has 4 negative factors; they pair up, so (−2)⁴ = +16. Statement II is false.
3. So only Statement I is true.

Teacher's Note:
1. Odd exponent on a negative base: negative answer. Even exponent: positive answer.
2. Watch the brackets: (−2)⁴ is not the same as −2⁴.

Answer: (b) Only Statement II is true

Explanation:
1. 4 + 4 + 4 = 12, which is 3 × 4, not 4³ = 64. Statement I is false.
2. 4 × 4 × 4 = 64 = 4³. Statement II is true.
3. So only Statement II is true.

Teacher's Note:
1. Repeated addition gives a product; repeated multiplication gives a power.
2. 4³ = 64, while 3 × 4 = 12.

Answer: (c) A is true, but R is false.

Explanation:
1. After 46 folds the thickness is 0.001 × 2⁴⁶ cm ≈ 7.04 × 10¹⁰ cm ≈ 7,03,687 km, which is more than 7,00,000 km. A is true.
2. Each fold doubles the thickness; it does not add a fixed 0.001 cm. R is false.
3. So A is true but R is false.

Teacher's Note:
1. Folding is multiplicative growth, not additive growth.
2. Adding 0.001 cm 46 times would give only 0.047 cm.

Answer: (d) A is false, but R is true.

Explanation:
1. 0⁰ would come from 0^a ÷ 0^a = 0 ÷ 0, which is not defined. So A is false.
2. R is true: the division rule needs n ≠ 0.
3. So A is false but R is true.

Teacher's Note:
1. n⁰ = 1 holds only for a non-zero base.
2. Division by 0 is never allowed.

Answer: (b) (i)-(s), (ii)-(p), (iii)-(q), (iv)-(r)

Explanation:
1. 2⁵ = 32: (i)-(s).
2. 5³ = 125: (ii)-(p).
3. 4² = 16: (iii)-(q).
4. 3⁴ = 81: (iv)-(r). So the answer is (b).

Teacher's Note:
1. Work out each power first, then match.
2. Only option (b) has (i)-(s) and (ii)-(p) together.

Answer: (d) 10th fold

Explanation:
1. Thickness after n folds = 0.001 × 2ⁿ cm.
2. After the 9th fold it is 0.001 × 512 = 0.512 cm, still below 1 cm.
3. After the 10th fold it is 0.001 × 1024 = 1.024 cm, the first bar above the 1 cm line. So the answer is the 10th fold.

Teacher's Note:
1. Look for the first bar that crosses the red dashed line.
2. 2¹⁰ = 1024 is worth remembering.

Answer: (c) 4² × 4³

Explanation:
1. (4²)³ = 4^2×3 = 4⁶ and (4³)² = 4^3×2 = 4⁶.
2. 4⁴ × 4² = 4⁴⁺² = 4⁶.
3. But 4² × 4³ = 4²⁺³ = 4⁵, so (c) is not equal to 4⁶.

Teacher's Note:
1. Power of a power: multiply exponents. Product: add exponents.
2. Check each option before choosing.

Answer: (a) 5.07 × 10⁷

Explanation:
1. 5,07,00,000 = 50700000 has 8 digits, so the decimal point moves 7 places to the left.
2. 5,07,00,000 = 5.07 × 10⁷.
3. 50.7 × 10⁶ and 0.507 × 10⁸ have the same value, but their coefficients are not between 1 and 10, so they are not in standard form.

Teacher's Note:
1. In standard form the coefficient must satisfy 1 ≤ x < 10.
2. Count the places the decimal point moves.

Answer: (d) 10¹¹

Explanation:
1. 1 crore = 10⁷, and a hundred crores make 1 arab = 10⁹.
2. A hundred arabs make 1 kharab = 10² × 10⁹ = 10¹¹.

Teacher's Note:
1. Lakh, crore, arab, kharab, neel: each is 100 times the previous one.
2. So the exponents go 5, 7, 9, 11, 13.

Answer: (a) 25

Explanation:
1. Same base, so add and subtract the exponents: −3 + 7 − 2 = 2.
2. 5⁻³ × 5⁷ ÷ 5² = 5² = 25.

Teacher's Note:
1. The laws work for negative exponents too.
2. Handle the signs of the exponents carefully.

Answer: (b) 26⁴

Explanation:
1. Each of the 4 slots can be filled in 26 ways, independently of the others.
2. Total = 26 × 26 × 26 × 26 = 26⁴.

Teacher's Note:
1. The number of choices per slot is the base; the number of slots is the exponent.
2. 26⁴ = 4,56,976 codes.

Answer: (c) Both statements are true

Explanation:
1. Using (n^a)^b = n^ab: (2⁵)² = 2¹⁰, so Statement I is true.
2. (2²)⁵ = 2¹⁰ as well, since 2 × 5 = 5 × 2 = 10. Statement II is true.
3. So both statements are true.

Teacher's Note:
1. The order of the two exponents does not matter in a power of a power.
2. Check: 32² = 1024 and 4⁵ = 1024.

Answer: (b) Only Statement II is true

Explanation:
1. In standard form the coefficient x must satisfy 1 ≤ x < 10.
2. 59 is not less than 10, so 59 × 10² is not standard form. Statement I is false.
3. 5900 = 5.9 × 10³ is correct standard form. So only Statement II is true.

Teacher's Note:
1. A correct value is not enough; the form must also be right.
2. Exactly one non-zero digit comes before the decimal point.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 2⁴ × 3⁴ = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) = 6⁴ = 1296. A is true.
2. R is the general rule for powers with the same exponent, and it is true.
3. A is just R with m = 2, n = 3, a = 4, so R correctly explains A.

Teacher's Note:
1. Same exponent: multiply the bases and keep the exponent.
2. Check: 16 × 81 = 1296 = 6⁴.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. The exponent 6 is even, so the six negative signs pair up and (−2)⁶ = +64. A is true.
2. (−2)⁶ = 64 and 64 = 4³, so R is true.
3. But being a perfect cube has nothing to do with the sign; A holds because the exponent is even. So R is not the correct explanation of A.

Teacher's Note:
1. Ask whether the reason actually gives the cause of the assertion.
2. The sign of a power of a negative base depends on whether the exponent is odd or even.

Answer: (d) A is false, but R is true.

Explanation:
1. 10⁻⁴ = \( \frac{1}{10000} \) = 0.0001 and 10⁻³ = \( \frac{1}{1000} \) = 0.001.
2. 0.0001 < 0.001, so A is false.
3. R is the correct meaning of a negative exponent, so R is true. So A is false but R is true.

Teacher's Note:
1. With negative exponents, the more negative the exponent, the smaller the number.
2. Write both as decimals to compare.

Answer: (d) (i)-(r), (ii)-(s), (iii)-(q), (iv)-(p)

Explanation:
1. 3⁻² = \( \frac{1}{9} \): (i)-(r).
2. 10⁻³ = \( \frac{1}{1000} \) = 0.001: (ii)-(s).
3. (−2)⁻² = \( \frac{1}{(-2)^{2}} \) = \( \frac{1}{4} \): (iii)-(q).
4. 7⁰ = 1: (iv)-(p). So the answer is (d).

Teacher's Note:
1. (−2)⁻² is positive because the exponent is even.
2. Any non-zero number to the power 0 is 1.

Answer: (a) \( \frac{1}{9} \)

Explanation:
1. The entry before P is 3⁻¹ = \( \frac{1}{3} \).
2. One more step to the right divides by 3: \( \frac{1}{3} \) ÷ 3 = \( \frac{1}{9} \).
3. So P is 3⁻² = \( \frac{1}{9} \).

Teacher's Note:
1. Moving right lowers the exponent by 1.
2. A negative exponent gives a small positive fraction, never a negative number.

Answer: (d) 9

Explanation:
1. 9 = 3², so 9⁷ = (3²)⁷ = 3¹⁴.
2. 3²⁰ ÷ 3¹⁴ = 3⁶ = 729.
3. The units digit is 9.

Teacher's Note:
1. Bring both powers to the same base first.
2. 3⁶ = 729 is small enough to work out fully.

Answer: (b) 3²⁰

Explanation:
1. 2³⁰ = 4¹⁵ = 1,07,37,41,824.
2. 5¹² = 24,41,40,625.
3. 3²⁰ = 3,48,67,84,401, the largest. So 3²⁰ is the greatest.

Teacher's Note:
1. A bigger base does not always give a bigger power.
2. Writing 4¹⁵ as (2²)¹⁵ = 2³⁰ saves work.

Answer: (c) Both statements are true

Explanation:
1. (7⁻³)⁻² = 7^{(−3)×(−2)} = 7⁶. Statement I is true.
2. \( p^{4}q^{10}(pq)^{6} = p^{4}q^{10} \times p^{6}q^{6} = p^{10}q^{16} \). Statement II is true.
3. So both statements are true.

Teacher's Note:
1. (−3) × (−2) = +6.
2. Open (pq)⁶ first, then add exponents of the same letter.

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. A is true: in the prime factorisation of a square every exponent is a multiple of 2, and in a cube every exponent is a multiple of 3. If both hold, every exponent is a multiple of 6, so the number is a sixth power (for example 64 = 8² = 4³ = 2⁶).
2. R is true: (n³)² = (n²)³ = n⁶.
3. But R only shows that every sixth power is a square and a cube, which is the reverse of A. So both are true, but R is not the correct explanation of A.

Teacher's Note:
1. A statement and its converse are different claims.
2. The real reason for A is the prime-factor exponents being multiples of both 2 and 3.

Answer: (c) (i)-(r), (ii)-(p), (iii)-(s), (iv)-(q)

Explanation:
1. 2³ × 2⁻⁵ = 2⁻²: (i)-(r).
2. (2²)³ ÷ 2⁶ = 2⁶⁻⁶ = 2⁰: (ii)-(p).
3. 2⁻⁴ × 4³ = 2⁻⁴ × 2⁶ = 2²: (iii)-(s).
4. 8² ÷ 2⁻² = 2⁶ ÷ 2⁻² = 2⁶⁺² = 2⁸: (iv)-(q). So the answer is (c).

Teacher's Note:
1. Write 4 as 2² and 8 as 2³ first.
2. Dividing by a negative power adds to the exponent.

Answer: (a) 3.75 × 10²

Explanation:
1. Trees per human = \( \frac{3 \times 10^{12}}{8 \times 10^{9}} \).
2. Divide the coefficients and subtract the exponents: 0.375 × 10³.
3. In standard form this is 3.75 × 10², about 375 trees for each person.

Teacher's Note:
1. Divide coefficients, subtract exponents.
2. Re-adjust the coefficient so that it lies between 1 and 10.