Q1: What is the exponential form of 81 when expressed as a power of 3? (a) 3³ (b) 3⁴ (c) 3⁵ (d) 3²
Solution:
Ans: (b) Since 81 = 3 × 3 × 3 × 3, it can be written as 3⁴ in exponential form with base 3.
Q2: In the expression 10³, what is the base? (a) 3 (b) 1000 (c) 10 (d) 30
Solution:
Ans: (c) In 10³, the number 10 is the base, which represents the number being multiplied, and 3 is the exponent showing repetition.
Q3: What is the result of (-2)⁴? (a) -8 (b) 8 (c) -16 (d) 16
Solution:
Ans: (d) When a negative number is raised to an even power, the result is positive. Thus, (-2)⁴ = 16.
Q4: Which of the following is the prime factorisation of 72 using exponents? (a) 2² × 3³ (b) 2³ × 3² (c) 2⁴ × 3 (d) 2 × 3⁴
Solution:
Ans: (b) The prime factorisation of 72 is 2 × 2 × 2 × 3 × 3, which equals 2³ × 3².
Q5: Using the law of exponents, what is 5⁶ ÷ 5² equal to? (a) 5³ (b) 5⁸ (c) 5⁴ (d) 5¹²
Solution:
Ans: (c) When dividing powers with the same base, subtract the exponents: 5⁶ ÷ 5² = 5⁶⁻² = 5⁴.
Fill in the Blanks
Q1: The special name for 10² is 10 _____.
Solution:
Ans: squared
Q2: In exponential form, 256 can be written as 2 raised to the power _____.
Solution:
Ans: 8
Q3: The prime factorisation of 1000 using exponents is 2³ × _____.
Solution:
Ans: 5³
Q4: According to the law of exponents, aᵐ × aⁿ = _____.
Solution:
Ans: aᵐ⁺ⁿ
Q5: The speed of light in vacuum expressed in standard form is 3.0 × _____ m/s.
Solution:
Ans: 10⁸
True or False
Q1: In the expression 8³, the number 3 is called the base.
Solution:
Ans: False In 8³, the number 8 is the base and 3 is the exponent showing how many times the base is multiplied.
Q2: The value of (-2)³ is -8.
Solution:
Ans: True When a negative number is raised to an odd power, the result is negative. Therefore, (-2)³ equals -8.
Q3: The expressions a³b² and a²b³ are the same.
Solution:
Ans: False In a³b² and a²b³, the powers of a and b are different, making these two expressions different from each other.
Q4: Using exponents, 432 can be expressed as 2⁴ × 3³.
Solution:
Ans: True The prime factorisation of 432 is 2 × 2 × 2 × 2 × 3 × 3 × 3, which equals 2⁴ × 3³.
Q5: According to exponent rules, 2³ × 3³ can be written as (2 × 3)³.
Solution:
Ans: True When multiplying powers with the same exponents, aᵐ × bᵐ = (ab)ᵐ. Therefore, 2³ × 3³ = (2 × 3)³.
Match the Following
Column A
Column B
1. 10³
A. 2⁷ × 5³
2. Prime factorisation of 16,000
B. aᵐ⁺ⁿ
3. aᵐ × aⁿ
C. 125
4. 5³
D. 10 cubed
5. (aᵐ)ⁿ
E. aᵐⁿ
Solution:
Ans:
1 - D: The special name for 10³ is "10 cubed" as per the naming convention for powers of 10.
2 - A: The prime factorisation of 16,000 is 2⁷ × 5³ as shown in the examples provided in the content.
3 - B: The law of exponents states that when multiplying powers with the same base, we add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
4 - C: The value of 5³ is 5 × 5 × 5, which equals 125, as mentioned in the examples.
5 - E: The law of taking power of a power states that (aᵐ)ⁿ = aᵐⁿ, where we multiply the exponents together.
Short Answer Questions
Q1: Explain why exponents are useful when dealing with very large numbers. Give an example from real life.
Solution:
Ans: Exponents help express very large numbers in a compact and manageable form, making them easier to read, understand, and compare. For example, the Earth's mass is approximately 5,970,000,000,000,000,000,000,000 kg, which is difficult to comprehend. Using exponents, this can be simplified and expressed more clearly. Similarly, distances between planets and stars can be expressed using powers of 10, making calculations and comparisons much simpler for scientists and students.
Q2: What is the difference between base and exponent? Explain with an example.
Solution:
Ans: In an exponential expression, the base represents the number being multiplied, whilst the exponent indicates how many times the base is multiplied by itself. For example, in 10³, the number 10 is the base and 3 is the exponent. This means 10 is multiplied by itself 3 times: 10 × 10 × 10 = 1000. The base remains constant whilst the exponent determines the number of repetitions in the multiplication.
Q3: How do negative bases affect the result when raised to different powers? Explain with examples.
Solution:
Ans: When a negative number is raised to an odd power, the result is negative, whilst raising it to an even power gives a positive result. For example, (-2)³ = (-2) × (-2) × (-2) = -8, which is negative. However, (-2)⁴ = 16, which is positive because multiplying an even number of negative factors produces a positive result. This pattern helps predict the sign of the answer.
Q4: Explain the law of exponents for multiplying powers with the same base and provide an example.
Solution:
Ans: When multiplying powers with the same base, we add the exponents whilst keeping the base unchanged. The rule is aᵐ × aⁿ = aᵐ⁺ⁿ for any non-zero integer a and whole numbers m and n. For example, 2² × 2³ = 2²⁺³ = 2⁵ = 32. This law simplifies calculations involving repeated multiplication and makes working with large numbers much easier and more efficient.
Q5: What is standard form and why is it useful for expressing very large numbers?
Solution:
Ans: Standard form, also called scientific notation, expresses any number as a decimal between 1.0 and 10.0 multiplied by a power of 10. This form makes very large numbers easier to read and compare. For example, the speed of light in vacuum is 300,000,000 m/s, which can be written as 3.0 × 10⁸ m/s in standard form. This compact representation simplifies understanding and calculations involving extremely large values.
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