Q1
1 Mark
What is the value of 2^3 × 2^2?
A2^5
B2^6
C2^4
D2^7
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Correct answer: Option 1 — 2^5
Chapter 5 · Class 7 Mathematics
Exponents and Powers — Important Questions
SUMMARY: This chapter introduces the concepts of exponents and powers, explaining their properties and applications in mathematical calculations.
KEY TOPICS: exponents, powers, laws of exponents, multiplication of powers, division of powers, power of a power, negative exponents, standard form, scientific notation, comparison of numbers using exponents
Multiple Choice Questions (MCQs)
5 questions
Q2
1 Mark
Which of the following is the correct expression for (3^2)^3?
A3^5
B3^6
C3^9
D3^8
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Correct answer: Option 3 — 3^9
Q3
1 Mark
What is the value of 5^-2?
A1/25
B25
C1/5
D5
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Correct answer: Option 1 — 1/25
Q4
1 Mark
If 10^x = 1000, what is the value of x?
A2
B3
C4
D1
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Correct answer: Option 2 — 3
Q5
1 Mark
Which of the following numbers is in standard form?
A0.00045
B4.5 × 10^-4
C45 × 10^-3
D450 × 10^-5
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Correct answer: Option 2 — 4.5 × 10^-4
Short Answer Questions
5 questions
Q6
3 Marks
What is the value of 2^3 and explain how you arrived at that value?
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The value of 2^3 is 8. This is calculated by multiplying 2 by itself three times: 2 × 2 × 2 = 8.
Q7
3 Marks
State the law of exponents for multiplying powers with the same base and provide an example.
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The law of exponents for multiplying powers with the same base states that a^m × a^n = a^(m+n). For example, 3^2 × 3^3 = 3^(2+3) = 3^5 = 243.
Q8
3 Marks
How do you express 0.00056 in standard form?
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To express 0.00056 in standard form, we write it as 5.6 × 10^(-4). This is done by moving the decimal point four places to the right, which gives us a negative exponent.
Q9
3 Marks
What is the result of (5^2)^3 and explain the power of a power rule?
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The result of (5^2)^3 is 5^(2×3) = 5^6 = 15625. The power of a power rule states that when raising a power to another power, you multiply the exponents.
Q10
3 Marks
Explain what negative exponents represent with an example.
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Negative exponents represent the reciprocal of the base raised to the opposite positive exponent. For example, 2^(-3) = 1/(2^3) = 1/8.
Long Answer Questions
5 questions
Q11
6 Marks
Explain the laws of exponents with examples. Include at least three different laws and demonstrate how they can be applied in calculations.
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The laws of exponents are fundamental rules that govern the operations involving powers. The first law states that when multiplying two powers with the same base, you add the exponents: a^m × a^n = a^(m+n). For example, 2^3 × 2^2 = 2^(3+2) = 2^5 = 32. The second law states that when dividing two powers with the same base, you subtract the exponents: a^m ÷ a^n = a^(m-n). For instance, 5^4 ÷ 5^2 = 5^(4-2) = 5^2 = 25. The third law states that when raising a power to another power, you multiply the exponents: (a^m)^n = a^(m*n). For example, (3^2)^3 = 3^(2*3) = 3^6 = 729. These laws simplify calculations and are essential for working with exponents.
Q12
6 Marks
Define negative exponents and provide examples to illustrate their meaning. How can negative exponents be converted to positive exponents?
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Negative exponents represent the reciprocal of the base raised to the opposite positive exponent. For example, a^(-n) = 1/(a^n). If we take 2^(-3), it can be rewritten as 1/(2^3) = 1/8. This shows that negative exponents allow us to express very small numbers in a manageable form. Another example is 5^(-2), which equals 1/(5^2) = 1/25. Converting negative exponents to positive ones is crucial in simplifying expressions and understanding their values in calculations.
Q13
6 Marks
What is scientific notation, and how is it used to express very large or very small numbers? Provide an example of converting a large number into scientific notation.
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Scientific notation is a way of expressing numbers that are too large or too small in a more compact form. It is written as the product of a number between 1 and 10 and a power of 10. For example, the number 45000 can be expressed in scientific notation as 4.5 × 10^4. This is done by moving the decimal point four places to the left, which indicates that the original number is multiplied by 10 raised to the power of 4. Scientific notation simplifies calculations, especially in scientific fields where such numbers are common, making it easier to perform operations like multiplication and division.
Q14
6 Marks
Discuss the process of comparing numbers using exponents. How can exponents help in determining which of two numbers is greater? Provide an example.
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Comparing numbers using exponents involves evaluating the powers and their bases. When two numbers are expressed as powers, the base and the exponent both play a crucial role in determining which number is greater. For example, to compare 2^5 and 3^3, we first calculate their values: 2^5 = 32 and 3^3 = 27. Since 32 is greater than 27, we conclude that 2^5 > 3^3. In cases where the bases are the same, the number with the larger exponent is greater. Conversely, if the bases differ, we may need to calculate their values or use logarithms for comparison. This method is particularly useful when dealing with large numbers.
Q15
6 Marks
Explain the multiplication and division of powers with examples. How do these operations differ when dealing with the same base versus different bases?
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The multiplication and division of powers follow specific rules based on whether the bases are the same. When multiplying powers with the same base, the exponents are added: a^m × a^n = a^(m+n). For example, 4^2 × 4^3 = 4^(2+3) = 4^5 = 1024. In contrast, when dividing powers with the same base, the exponents are subtracted: a^m ÷ a^n = a^(m-n). For instance, 6^5 ÷ 6^2 = 6^(5-2) = 6^3 = 216. However, if the bases are different, such as 2^3 and 3^2, we cannot directly apply these laws and must calculate their values separately: 2^3 = 8 and 3^2 = 9. Thus, 8 and 9 can be compared directly, but the laws of exponents do not apply.
Assertion–Reason Questions
5 questions
Q16
1 Mark
Assertion (A): The expression 2^3 × 2^2 can be simplified to 2^5.
Reason (R): According to the laws of exponents, when multiplying powers with the same base, we add the exponents.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q17
1 Mark
Assertion (A): The expression 5^0 is equal to 1.
Reason (R): Any non-zero number raised to the power of zero is equal to zero.
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Correct answer: Option 3 —
A is true, but R is false.
Q18
1 Mark
Assertion (A): Negative exponents represent the reciprocal of the base raised to the positive exponent.
Reason (R): This is a fundamental property of exponents.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q19
1 Mark
Assertion (A): The expression (3^2)^3 equals 3^5.
Reason (R): When raising a power to another power, we multiply the exponents.
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Correct answer: Option 3 —
A is true, but R is false.
Q20
1 Mark
Assertion (A): The scientific notation of 4500 is 4.5 × 10^3.
Reason (R): Scientific notation requires the first term to be between 1 and 10.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Statement-Based Questions
5 questions
Q21
1 Mark
Statement 1: The expression 2^3 equals 8.
Statement 2: The expression 3^2 equals 9.
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Correct answer: Option 1 —
Both statements are true.
Q22
1 Mark
Statement 1: The law of exponents states that a^m × a^n = a^(m+n).
Statement 2: The law of exponents states that a^m ÷ a^n = a^(m-n).
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Correct answer: Option 1 —
Both statements are true.
Q23
1 Mark
Statement 1: The expression (2^3)^2 equals 2^6.
Statement 2: The expression (3^2)^3 equals 3^5.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q24
1 Mark
Statement 1: Negative exponents indicate the reciprocal of the base raised to the positive exponent.
Statement 2: The standard form of a number is always written with a base greater than 10.
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Correct answer: Option 2 —
Only Statement 1 is true.
Q25
1 Mark
Statement 1: In scientific notation, 0.0045 can be expressed as 4.5 × 10^-3.
Statement 2: In scientific notation, 1000 can be expressed as 1 × 10^3.
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Correct answer: Option 1 —
Both statements are true.
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