CBSE Class 8 Mathematics Power Play Practice Paper Assignment Set 01 with Answers

Class 8 Maths Ganita Prakash Chapter 2 Power Play Practice Paper 08 is a 40-mark test of 20 questions in Sections A to E, given here with step-by-step answers. It covers the zero exponent, negative exponents, the laws of exponents for multiplying, dividing and raising powers, solving equations such as 3ˣ = 9² × 27, and writing very large and very small numbers like the Sun–Mars distance and a virus's DNA length in standard form. The two case studies build powers from a king's gold coins and a doubling virus.

Section A: Multiple Choice Questions (1 Mark Each)

Section B: Short Answer Questions (2 Marks Each)

Question 11. Find the value of (i) (3² × 3⁴) ÷ 3⁶   (ii) (5⁸ ÷ 5³) × 5²
Answer:
(i) Same base, so add exponents when multiplying: 3² × 3⁴ = 3²⁺⁴ = 3⁶.
Then 3⁶ ÷ 3⁶ = 3⁶⁻⁶ = 3⁰ = 1.
(ii) Subtract exponents when dividing: 5⁸ ÷ 5³ = 5⁸⁻³ = 5⁵.
Then 5⁵ × 5² = 5⁵⁺² = 5⁷ (= 78125).
In simple words: Add exponents when multiplying and subtract when dividing: (i) 1, (ii) 5⁷.

Teacher's Note:
1. Any number divided by itself is 1, which matches 3⁰ = 1.
2. Work inside the brackets first.

Question 12. Simplify: \( \left(\frac{1}{2}\right)^{-2} \) + \( \left(\frac{1}{3}\right)^{-2} \) + \( \left(\frac{1}{4}\right)^{-2} \)
Answer:
A negative exponent turns a fraction upside down: \( \left(\frac{1}{n}\right)^{-2} \) = n².
\( \left(\frac{1}{2}\right)^{-2} \) = 2² = 4.
\( \left(\frac{1}{3}\right)^{-2} \) = 3² = 9.
\( \left(\frac{1}{4}\right)^{-2} \) = 4² = 16.
Sum = 4 + 9 + 16 = 29.
In simple words: Each term becomes a square: 4 + 9 + 16 = 29.

Teacher's Note:
1. \( \left(\frac{a}{b}\right)^{-n} \) = \( \left(\frac{b}{a}\right)^{n} \).
2. Simplify each term before adding; there is no rule for adding powers.

Question 13. If \( 3^{x} = 9^{2} \times 27 \), find the value of x.
Answer:
Write every number as a power of 3: 9 = 3² and 27 = 3³.
9² = (3²)² = 3⁴.
So \( 3^{x} \) = 3⁴ × 3³ = 3⁴⁺³ = 3⁷.
Equal bases, so equal exponents: x = 7.
In simple words: 9² × 27 = 3⁴ × 3³ = 3⁷, so x = 7.

Teacher's Note:
1. Convert to a common base first.
2. Check: 3⁷ = 2187 and 81 × 27 = 2187.

Question 14. A country's GDP is Rs. 3.5 × 10¹². The population is 7 × 10⁸. What is the GDP per person? Express your answer in standard form.
Answer:
GDP per person = GDP ÷ population = (3.5 × 10¹²) ÷ (7 × 10⁸).
Divide the numbers: 3.5 ÷ 7 = 0.5.
Divide the powers: 10¹² ÷ 10⁸ = 10¹²⁻⁸ = 10⁴.
So the answer is 0.5 × 10⁴. In standard form 0.5 = 5 × 10⁻¹, so 0.5 × 10⁴ = 5 × 10³.
GDP per person = Rs. 5 × 10³ (Rs. 5000).
In simple words: Divide the numbers and the powers of 10 separately: Rs. 5 × 10³ per person.

Teacher's Note:
1. The number in front must be between 1 and 10 in standard form.
2. Adjust the power of 10 when you move the decimal point.

Section C: Short Answer Questions (3 Marks Each)

Question 15. Simplify: \( \frac{6^{3} \times 12^{2} \times 9}{18^{2} \times 24} \)
Answer:
Write each number using prime factors: 6 = 2 × 3, 12 = 2² × 3, 9 = 3², 18 = 2 × 3², 24 = 2³ × 3.
Numerator: 6³ × 12² × 9 = (2³ × 3³) × (2⁴ × 3²) × 3² = 2⁷ × 3⁷.
Denominator: 18² × 24 = (2² × 3⁴) × (2³ × 3) = 2⁵ × 3⁵.
Divide: 2⁷⁻⁵ × 3⁷⁻⁵ = 2² × 3² = 4 × 9 = 36.
In simple words: Break every number into powers of 2 and 3 and cancel: the answer is 36.

Teacher's Note:
1. Prime factorisation makes cancelling easy.
2. Collect powers of 2 and powers of 3 separately.

Question 16. Find the value of x for which \( \left(\frac{5}{7}\right)^{2x-1} \) ÷ \( \left(\frac{5}{7}\right)^{3} \) = \( \left(\frac{5}{7}\right)^{-4} \)
Answer:
All terms have the same base \( \frac{5}{7} \), so subtract exponents when dividing.
Left side = \( \left(\frac{5}{7}\right)^{(2x-1)-3} \) = \( \left(\frac{5}{7}\right)^{2x-4} \).
Equate exponents: 2x − 4 = −4.
2x = 0, so x = 0.
Check: \( \left(\frac{5}{7}\right)^{-1} \) ÷ \( \left(\frac{5}{7}\right)^{3} \) = \( \left(\frac{5}{7}\right)^{-4} \). ✓
In simple words: Same base on both sides, so 2x − 4 = −4 and x = 0.

Teacher's Note:
1. With equal bases, equate the exponents.
2. x = 0 is a perfectly valid answer.

Question 17. By what number should \( \left(\frac{-3}{4}\right)^{-2} \) be multiplied so that the product is 3⁻²?
Answer:
Let the number be x. Then x × \( \left(\frac{-3}{4}\right)^{-2} \) = 3⁻².
\( \left(\frac{-3}{4}\right)^{-2} \) = \( \left(\frac{4}{-3}\right)^{2} \) = \( \frac{16}{9} \).
3⁻² = \( \frac{1}{9} \).
So x = \( \frac{1}{9} \) ÷ \( \frac{16}{9} \) = \( \frac{1}{9} \) × \( \frac{9}{16} \) = \( \frac{1}{16} \).
In simple words: Divide 3⁻² = 1/9 by 16/9 to get \( \frac{1}{16} \).

Teacher's Note:
1. To find the missing factor, divide the product by the given factor.
2. An even power removes the minus sign.

Section D: Long Answer Question (5 Marks)

Question 18. Find the value of x and y such that \( 5^{x} = 125 \) and \( 7^{y} = \frac{1}{343} \). Hence, evaluate the expression \( (x - y)^{x + y} \).
Answer:
Finding x: 125 = 5 × 5 × 5 = 5³, so 5ˣ = 5³ and x = 3.
Finding y: 343 = 7³, so \( \frac{1}{343} \) = \( \frac{1}{7^{3}} \) = 7⁻³.
So 7ʸ = 7⁻³ and y = −3.
Evaluating: x − y = 3 − (−3) = 6 and x + y = 3 + (−3) = 0.
\( (x - y)^{x + y} \) = 6⁰ = 1.
In simple words: x = 3, y = −3, and the expression is 6⁰ = 1.

Teacher's Note:
1. A fraction 1/nᵃ can be written as n⁻ᵃ.
2. Be careful with signs: 3 − (−3) = 6.
3. Any non-zero number to the power 0 is 1.

Section E: Case Study Based Questions (4 Marks Each)

Question 19. Case Study 1: The King's Gold Coins. A king distributes his gold coins to his three ministers. Each minister receives three large chests. Each chest contains three medium-sized boxes. Each medium-sized box contains three small pouches, and each small pouch contains three gold coins. The king asks his ministers to calculate the total number of gold coins, but they must use exponential notation to present their answer.
(i) Express the number of gold coins each minister receives in exponential form.
(ii) How many gold coins were distributed in total by the king? Express your answer in exponential form.
(iii) If the number of ministers was four instead of three, and the number of chests, boxes, pouches and coins per pouch remained the same, how many total coins would have been distributed? Express your answer in exponential form.
(iv) The king takes back a portion of the total coins equal to 3⁴. What is the number of coins remaining with the ministers? Express your answer in exponential form.
Answer:
(i) Each minister gets 3 chests × 3 boxes × 3 pouches × 3 coins = 3⁴ coins (= 81).
(ii) Total = 3 ministers × 3⁴ = 3¹⁺⁴ = 3⁵ coins (= 243).
(iii) With 4 ministers: 4 × 3⁴ = 2² × 3⁴ coins (= 324).
(iv) Remaining = 3⁵ − 3⁴ = 3 × 3⁴ − 1 × 3⁴ = (3 − 1) × 3⁴ = 2 × 3⁴ coins (= 162).
In simple words: Each level multiplies by 3, so one minister gets 3⁴ coins and the king gives out 3⁵ in all.

Teacher's Note:
1. Repeated multiplication by the same number gives a power.
2. 3⁵ − 3⁴ is not 3¹; take out the common factor 3⁴ instead.

Question 20. Case Study 2: The Virus Spread. A new virus strain is discovered that infects people at an alarming rate. Initially, there is only one person infected on Day 0. The number of infected people doubles every day. The government wants to track the spread of this virus using exponential notation to manage resources efficiently.
(i) Write the number of infected people at the end of Day 5 in exponential form.
(ii) On which day will the number of infected people reach 256?
(iii) If the number of infected people tripled every day instead of doubling, how many people would be infected at the end of Day 3? Express your answer in exponential form.
(iv) The number of infected people after x days can be represented by \( 2^{x} \). What is the ratio of the number of infected people on Day 8 to Day 4? Express in exponential form.
Answer:
Day 0 has 1 = 2⁰ person, Day 1 has 2¹, Day 2 has 2², and so on: after n days there are 2ⁿ people.
(i) End of Day 5: 2⁵ people (= 32).
(ii) 256 = 2⁸, so the number reaches 256 on Day 8.
(iii) Tripling from 1 person: after n days there are 3ⁿ people, so at the end of Day 3 there are 3³ people (= 27).
(iv) Ratio = 2⁸ : 2⁴ = 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ (= 16), that is 2⁴ : 1.
In simple words: Doubling from 1 gives 2ⁿ after n days, so Day 5 has 2⁵ and Day 8 has 256 = 2⁸ people.

Teacher's Note:
1. Starting from 1 on Day 0 makes the exponent equal to the day number.
2. A ratio of powers with the same base is found by subtracting exponents.