CBSE Class 8 Mathematics Exponents And Powers Worksheet Set 05

Question. 1. A gardner plants 17956 trees in such a way that there are as many rows as there are trees in each row. The number of trees in a row are :
(a) 136
(b) 164
(c) 134
(d) 166
Answer: C

Question. 2. \(\sqrt{2\sqrt{2\sqrt{2\sqrt{2\sqrt{2}}}}}\) is equal to
(a) 0
(b) 2
(c) 1
(d) \(2^{\frac{31}{32}}\)
Answer: D

Question. 3. 'The greatest six digit number which is perfect square is :
(a) 998004
(b) 998006
(c) 998049
(d) 998001
Answer: D

Question. 4. Divide the number 26244 by the smallest number so that the quotient is a perfect cube, so the smallest number is :
(a) 4
(b) 6
(c) 36
(d) 16
Answer: C

Question. 5. The perfect cube nearest to 2750 is :
(a) 2749
(b) 2747
(c) 2744
(d) 2754
Answer: C

Question. 6. The least number which must be added to 4931 to make it perfect square is :
(a) 110
(b) 120
(c) 130
(d) None
Answer: A

Question. 7. The least number which must be subtracted from 18265 to make it a perfect square is:
(a) 20
(b) 30
(c) 40
(d) 50
Answer: C

Question. 8. Least perfect square of 6 digits is :
(a) 998000
(b) 998001
(c) 998002
(d) None of these
Answer: B

Question. 9. A \(8 \times 6 \times 4 \text{ cm}^3\) metallic cube is melted. Find the minimum volume of molten metal which should be added to mould it into a cube whose edge is an x, where x is an integer.
(a) 20
(b) 21
(c) 23
(d) 24
Answer: D

Question. 10. Value of \(\sqrt[3]{392} \times \sqrt[3]{448}\) is
(a) 50
(b) 52
(c) 54
(d) 56
Answer: D

Question. 11. The smallest number by which 137592 should be multiply to make it perfect cube.
(a) 1183
(b) 1180
(c) 1181
(d) None of these
Answer: A

Question. 12. The volumes of two cubes are in the ratio of \(343 : 1331\), the ratio of their edges is .
(a) 7: 10
(b) 7:11
(c) 7: 12
(d) None of these
Answer: B

Question. 13. (5, 12, 15) and (7, 5, 12)
(a) Both are Pythagorean triplets
(b) None of the both is Pythagorean triplet
(c) First is Pythagorean triplet while second is not
(d) Second is Pythagorean triplet while first is not
Answer: B

Question. 14. \(\frac{(65 - 56)^2}{3^2} = ?\)
(a) 8
(b) 9
(c) \(\frac{1}{9}\)
(d) None of these
Answer: B

Question. 15. \((16)^2 \times (4)^2 = ?\)
(a) 4096
(b) 16360
(c) 25360
(d) None of these
Answer: A

Question. 16. \(\left( \frac{15}{8} \times 16 \right)^2 = ?\)
(a) 916
(b) 900
(c) 926
(d) 1966
Answer: B

Question. 17. \(\frac{(16)^2 + (12)^2}{(20)^2} = ?\)
(a) 4
(b) 2
(c) 1
(d) 5
Answer: C

Question. 18. \(\frac{(48 - 8)^2}{40} = ?\)
(a) 40
(b) 80
(c) 15
(d) 150
Answer: A

Question. 19. \((65)^2 - (64)^2 + (1)^2 =?\)
(a) 130
(b) 129
(c) 138
(d) 120
Answer: A

Question. 20. \((57)^2 - (52)^2 = 5 \times ?\)
(a) 109
(b) 108
(c) 5
(d) None of these
Answer: A

Question. 21. If \((65)^2 - (60)^2 = [(55)^2 - (50)^2] \times y\), then the value of y is-
(a) \(\frac{23}{21}\)
(b) \(\frac{28}{21}\)
(c) \(\frac{25}{21}\)
(d) None of these
Answer: C

Question. 22. \(\frac{55 \times 55 - 2 \times 25 \times 55 + 25 \times 25}{30^2} = ?\)
(a) 6
(b) 4
(c) 5
(d) None of these
Answer: D

Question. 23. \(\frac{65 \times 65 + 2 \times 65 \times 35 + 35 \times 35}{65 + 35} = ?\)
(a) 1000
(b) 10
(c) 100
(d) None of these
Answer: C

Question. 24. By studying the numbers given below, find out the missing numbers.
\((1)^2 + (2)^2 + (2)^2 = (3)^2\)
(i) \((4)^2 + (5)^2 + \square^2 = (21)^2\)
(a) 20
(b) 6
(c) 7
(d) 24
(ii) \((5)^2 + \square^2 + (30)^2 = (31)^2\)
(a) 9
(b) 8
(c) 7
(d) 6
Answer: (i) a; (ii) d

Question. 25. The value of \(\sqrt{225}\) is-
(a) 25
(b) 15
(c) 35
(d) 05
Answer: B

Choose the correct option (Question No. 1 - 5)

Question 1. \(\left(\frac{1}{2}\right)^4 \times \left(\frac{2}{3}\right)^3\) is
(a) \(\frac{2^3}{6^7}\)
(b) \(\left(\frac{2}{6}\right)^{12}\)
(c) \(\left(\frac{2}{5}\right)^7\)
(d) \(\frac{1}{2 \times 3^3}\)
Answer: (d) \(\frac{1}{2 \times 3^3}\)
\(\left(\frac{1}{2}\right)^4 \times \left(\frac{2}{3}\right)^3 = \frac{1^4}{2^4} \times \frac{2^3}{3^3}\)
\(= \frac{1}{2^4 \times 2^{-3} \times 3^3} = \frac{1}{2^{4-3} \times 3^3} = \frac{1}{2^1 \times 3^3} = \frac{1}{2 \times 3^3}\)
In simple words: Expand powers and cancel common terms in numerator and denominator.

Exam Tip: Use laws of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to simplify expressions quickly.

Question 2. Multiplicative inverse of \(5^{-3}\) is
(a) \(\left(\frac{1}{5}\right)^{-3}\)
(b) \(5^3\)
(c) \(\left(-\frac{1}{5}\right)^{-3}\)
(d) \(-5^{-3}\)
Answer: (b) \(5^3\)
The multiplicative inverse of a number \(a\) is \(\frac{1}{a}\).
The multiplicative inverse of \(5^{-3}\) is \(\frac{1}{5^{-3}} = 5^3\).
In simple words: The multiplicative inverse changes the sign of the exponent.

Exam Tip: Remember that \(a^{-m} = \frac{1}{a^m}\), so its reciprocal is \(a^m\).

Question 3. Which of the following is true
(a) \(15^0 < 7^0\)
(b) \(15^0 < 7^0\)
(c) \(15^0 = 7^0\)
(d) \(15^0 = 7\)
Answer: (c) \(15^0 = 7^0\)
Any non-zero number raised to the power 0 is equal to 1.
Therefore, \(15^0 = 1\) and \(7^0 = 1\), which implies \(15^0 = 7^0\).
In simple words: Zero exponent rule states that anything to the power 0 is 1.

Exam Tip: Always check if the base is non-zero before applying the zero exponent rule.

Question 4. Which of the following is a false statement?
(a) \(a^m \div a^n = a^{m-n}\)
(b) \(a^m \times b^m = ab^m\)
(c) \(a^{-m} \times a^n = a^{-m-n}\)
(d) \(a^{-m} = a^{-mn}\)
Answer: (d) \(a^{-m} = a^{-mn}\)
Statement (d) is incorrect because there is no exponent rule stating that \(a^{-m} = a^{-mn}\). The correct power of a power rule is \((a^m)^n = a^{mn}\).
In simple words: Look for standard algebraic identities that have been incorrectly written.

Exam Tip: Memorize all standard laws of exponents to easily spot false statements.

Question 5. \(4^{-3}\) in exponential form with base 2 is
(a) \(2^6\)
(b) \(2^3\)
(c) \((-2)^6\)
(d) \(2^{-6}\)
Answer: (d) \(2^{-6}\)
\(4^{-3} = (2^2)^{-3} = 2^{2 \times (-3)} = 2^{-6}\)
In simple words: Express base 4 as \(2^2\) and apply the power of a power rule.

Exam Tip: Convert composite bases to prime bases before applying exponent laws.

Question 6. (a) Express with positive exponents
(i) \(3^{-5}\)
(ii) \(5^{-6}\)
(iii) \(\left(\frac{-6}{5}\right)^{-11}\)
Answer:
(i) \(\frac{1}{3^5}\)
(ii) \(\frac{1}{5^6}\)
(iii) \(\left(\frac{-5}{6}\right)^{11}\)
In simple words: Take the reciprocal of the base and make the exponent positive.

Exam Tip: Use \(a^{-m} = \left(\frac{1}{a}\right)^m\) to convert negative exponents to positive.

Question 6. (b) In the similar manner, we can also convert negative attitude into positive attitude by using (i) Body (ii) Mind (iii) Intellect (iv) Gun. Choose the correct option
Answer: (iii) Intellect
A higher level of intellect and reasoning helps transform negative thinking into positive constructive attitudes.
In simple words: Higher wisdom (intellect) converts negativity into positivity.

Exam Tip: Value-based questions assess moral reasoning and logical application.

Question 7. Find the value.
(a) \(-3^{-2}\)
(b) \(4^{-2}\)
(c) \(100^{-1}\)
(d) \(-1^{-27}\)
(e) \(\left(\frac{1}{3}\right)^{-5}\)
Answer:
(a) \(-3^{-2} = -\frac{1}{3^2} = -\frac{1}{9}\)
(b) \(4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
(c) \(100^{-1} = \frac{1}{100}\)
(d) \(-1^{-27} = -\frac{1}{1^{27}} = -1\)
(e) \(\left(\frac{1}{3}\right)^{-5} = 3^5 = 243\)
In simple words: Apply negative exponent rules to compute the final fractions or whole numbers.

Exam Tip: Pay careful attention to parentheses when negative signs are present with exponents.

Question 8. Express as a rational number.
(a) \(5^{-1}\)
(b) \(5^{-2}\)
(c) \(\left(-\frac{3}{4}\right)^{-3}\)
(d) \(4^3 \times 4^{-5}\)
(e) \(\left[\left(\frac{3}{2}\right)^{-2}\right]^2\)
Answer:
(a) \(5^{-1} = \frac{1}{5}\)
(b) \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)
(c) \(\left(-\frac{3}{4}\right)^{-3} = \left(-\frac{4}{3}\right)^3 = -\frac{64}{27}\)
(d) \(4^3 \times 4^{-5} = 4^{3-5} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16}\)
(e) \(\left[\left(\frac{3}{2}\right)^{-2}\right]^2 = \left(\frac{3}{2}\right)^{-4} = \left(\frac{2}{3}\right)^4 = \frac{16}{81}\)
In simple words: Simplify using exponent laws and express the result as a fraction.

Question 9. Evaluate
(a) \(3^{-7} \div 3^{-10} \times 3^{-5}\)
(b) \(3^0 - 4^0 + 3^0\)
(c) \((4^{-1} \times 3^1) \div 6^{-1}\)
(d) \(\left(\frac{2}{3}\right)^{-3} \times \left(\frac{2}{3}\right)^{-2} \div \left(\frac{2}{3}\right)^5\)
(e) \(\left(\frac{-5}{8}\right)^7 \div \left(\frac{-5}{8}\right)^5\)
Answer:
(a) \(3^{-7-(-10)+(-5)} = 3^{-7+10-5} = 3^{-2} = \frac{1}{9}\)
(b) \(1 - 1 + 1 = 1\)
(c) \(\left(\frac{1}{4} \times 3\right) \div \frac{1}{6} = \frac{3}{4} \times 6 = \frac{9}{2}\)
(d) \(\left(\frac{2}{3}\right)^{-3+(-2)-5} = \left(\frac{2}{3}\right)^{-10} = \left(\frac{3}{2}\right)^{10}\)
(e) \(\left(\frac{-5}{8}\right)^{7-5} = \left(\frac{-5}{8}\right)^2 = \frac{25}{64}\)
In simple words: Combine exponents with the same base using addition and subtraction rules.

Exam Tip: Follow the order of operations and combine like bases step-by-step.

Question 10. Simplify and write the answer with positive exponents.
(a) \((5^{-1} \div 3^{-1})^2\)
(b) \((4^{-1} - 7^{-1})^{-1} \div (3^{-1} + 7^{-1})^{-1}\)
(c) \(\left[\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3}\right] \div \left(\frac{1}{4}\right)^{-2}\)
Answer:
(a) \(\left(\frac{1}{5} \div \frac{1}{3}\right)^2 = \left(\frac{3}{5}\right)^2 = \frac{9}{25}\)
(b) \(\left(\frac{1}{4} - \frac{1}{7}\right)^{-1} \div \left(\frac{1}{3} + \frac{1}{7}\right)^{-1} = \left(\frac{3}{28}\right)^{-1} \div \left(\frac{10}{21}\right)^{-1} = \frac{28}{3} \times \frac{10}{21} = \text{undefined/recalculate...}\)
Let us re-evaluate (b):
\(\left(\frac{7-4}{28}\right)^{-1} = \left(\frac{3}{28}\right)^{-1} = \frac{28}{3}\)
\(\left(\frac{7+3}{21}\right)^{-1} = \left(\frac{10}{21}\right)^{-1} = \frac{21}{10}\)
\(\frac{28}{3} \div \frac{21}{10} = \frac{28}{3} \times \frac{10}{21} = \frac{4 \times 10}{3 \times 3} = \frac{40}{9}\)
(c) \([3^3 - 2^3] \div 4^2 = [27 - 8] \div 16 = \frac{19}{16}\)
In simple words: Simplify inner brackets into fractions before applying powers.

Exam Tip: Convert negative exponents to fractions first for easier addition and subtraction inside parentheses.

Question 11. Simplify.
(a) \(5^7 \times 3^3 \div 5^{10}\)
(b) \(-8^5 \div -8^5\)
(c) \(\left(\frac{3}{4}\right)^4 \times \left(\frac{3}{4}\right)^5 \div \left(\frac{-3}{4}\right)^9\)
(d) \(2^3 \times 5 \div 2^{15}\)
Answer:
(a) \(5^{7-10} \times 3^3 = 5^{-3} \times 3^3 = \frac{3^3}{5^3} = \frac{27}{125}\)
(b) \(-8^{5-5} = -8^0 = -1\)
(c) Note that \(\left(\frac{-3}{4}\right)^9 = -\left(\frac{3}{4}\right)^9\).
\(\left(\frac{3}{4}\right)^{4+5-9} \div (-1) = \left(\frac{3}{4}\right)^0 \div (-1) = 1 \div (-1) = -1\)
(d) Assuming \(3 \times 15^{22}\) or similar transcription, rewriting standard form: \(2^3 \times 5 \div 2^{15} = 5 \times 2^{3-15} = 5 \times 2^{-12} = \frac{5}{2^{12}}\)
In simple words: Group terms with identical bases and subtract exponents during division.

Exam Tip: Watch out for negative bases raised to odd powers, as they retain the negative sign.

Question 12. If \(x = \left(\frac{5}{4}\right)^5 \div \left(\frac{5}{4}\right)^3\) find \(x^2\)
Answer:
\(x = \left(\frac{5}{4}\right)^{5-3} = \left(\frac{5}{4}\right)^2 = \frac{25}{16}\)
\(x^2 = \left(\frac{25}{16}\right)^2 = \frac{625}{256}\)
In simple words: Find the value of x first by subtracting exponents, then square it.

Exam Tip: Simplify the expression for x completely before substituting into the target expression.

Question 13. Find 'x' if
(a) \(3^{x-2} \div 3^{-3} = 3^4\)
(b) \(5^{2x+1} = 125\)
(c) \((-2)^{x+1} \times (-2)^5 = (-2)^7\)
(d) \(\left(\frac{2}{5}\right)^3 \times \left(\frac{2}{5}\right)^{-6} = \left(\frac{2}{5}\right)^{2x-1}\)
Answer:
(a) \(3^{x-2-(-3)} = 3^4 \implies 3^{x+1} = 3^4 \implies x+1 = 4 \implies x = 3\)
(b) \(5^{2x+1} = 5^3 \implies 2x+1 = 3 \implies 2x = 2 \implies x = 1\)
(c) \((-2)^{x+1+5} = (-2)^7 \implies (-2)^{x+6} = (-2)^7 \implies x+6 = 7 \implies x = 1\)
(d) \(\left(\frac{2}{5}\right)^{3+(-6)} = \left(\frac{2}{5}\right)^{2x-1} \implies -3 = 2x-1 \implies 2x = -2 \implies x = -1\)
In simple words: Equate the exponents on both sides when the bases are equal.

Exam Tip: Base equality allows direct comparison of exponents: if \(a^m = a^n\), then \(m = n\).

Question 14. By what number should \(\left(\frac{-3}{2}\right)^{-1}\) be divided so that the quotient may be equal to \(\frac{1}{6}\).
Answer:
Let the required number be \(y\).
\(\left(\frac{-3}{2}\right)^{-1} \div y = \frac{1}{6}\)
\(-\frac{2}{3} \div y = \frac{1}{6}\)
\(y = -\frac{2}{3} \div \frac{1}{6} = -\frac{2}{3} \times 6 = -4\)
In simple words: Set up a division equation and solve for the unknown variable.

Exam Tip: Translate "divided by y" directly into algebraic equations to avoid confusion.

Question 15. By what number should
(a) \(7^4\) be multiplied so that the result is 7?
(b) \(\left(\frac{-2}{9}\right)^{-2}\) be divided to get 3?
Answer:
(a) Let the number be \(y\). \(7^4 \times y = 7^1 \implies y = \frac{7^1}{7^4} = 7^{1-4} = 7^{-3} = \frac{1}{343}\)
(b) Let the number be \(y\). \(\left(\frac{-2}{9}\right)^{-2} \div y = 3 \implies \left(\frac{9}{-2}\right)^2 \div 3 = y \implies \frac{81}{4} \div 3 = y \implies y = \frac{27}{4}\)
In simple words: Use multiplication and division principles with exponents to find the missing factor.

Exam Tip: Multiplication questions use division to find the missing factor, and vice versa.

Question 16. Simplify.
(a) \(\frac{5^2 \times p^{-4}}{5^3 \times 10 \times p^{-8}} \quad (p \neq 0)\)
(b) \(\frac{10^{-5} \times 125 \times 3^{-5}}{6^{-5} \times 5^7}\)
(c) \(\frac{27 \times x^{-2}}{3^{-2} \times x^{-8}} \quad (x \neq 0)\)
Answer:
(a) \(\frac{5^2 p^{-4}}{5^3 \times 2 \times 5 \times p^{-8}} = \frac{5^2 p^{-4}}{5^4 \times 2 \times p^{-8}} = \frac{p^{-4-(-8)}}{5^{4-2} \times 2} = \frac{p^4}{25 \times 2} = \frac{p^4}{50}\)
(b) Prime factorization: \(10 = 2 \times 5\), \(125 = 5^3\), \(6 = 2 \times 3\).
\(\frac{(2 \times 5)^{-5} \times 5^3 \times 3^{-5}}{(2 \times 3)^{-5} \times 5^7} = \frac{2^{-5} 5^{-5} 5^3 3^{-5}}{2^{-5} 3^{-5} 5^7} = \frac{5^{-2}}{5^7} = 5^{-9} = \frac{1}{5^9}\)
(c) \(\frac{3^3 \times x^{-2}}{3^{-2} \times x^{-8}} = 3^{3-(-2)} \times x^{-2-(-8)} = 3^5 \times x^6 = 243x^6\)
In simple words: Break numbers into prime factors and subtract exponents of matching bases.

Exam Tip: Always convert composite numbers like 6, 10, and 125 into prime power components.

Question 17. Express in standard form
(a) \(243 \times 10^5\)
(b) \(0.0009\)
(c) Fifty Thousand
Answer:
(a) \(2.43 \times 10^2 \times 10^5 = 2.43 \times 10^7\)
(b) \(9 \times 10^{-4}\)
(c) \(50,000 = 5 \times 10^4\)
In simple words: Write numbers as a decimal between 1 and 10 multiplied by a power of 10.

Exam Tip: Moving the decimal point to the left increases the exponent of 10, while moving it to the right decreases it.

Question 18. Express in usual form
(a) \(5.4 \times 10^3\)
(b) \(8 \times 10^{-2}\)
(c) \(5.08 \times 10^4\)
Answer:
(a) \(5.4 \times 1000 = 5400\)
(b) \(\frac{8}{100} = 0.08\)
(c) \(5.08 \times 10000 = 50800\)

In simple words: Multiply the decimal by the power of 10 to get the standard number.
Exam Tip: Positive powers of 10 move the decimal to the right; negative powers move it to the left.

Question 19. Simplify \(\frac{1}{1 + x^{n-m}} + \frac{1}{1 + x^{m-n}}\)
Answer:
\(\frac{1}{1 + \frac{x^n}{x^m}} + \frac{1}{1 + \frac{x^m}{x^n}} = \frac{1}{\frac{x^m + x^n}{x^m}} + \frac{1}{\frac{x^n + x^m}{x^n}}\)
\(= \frac{x^m}{x^m + x^n} + \frac{x^n}{x^m + x^n} = \frac{x^m + x^n}{x^m + x^n} = 1\)
In simple words: Convert subtraction in exponents into fractions and take the common denominator.

Exam Tip: This is a standard algebraic identity simplification yielding 1.

Question 20. \(6^{2x+1} \div 26 = 216\) (Find the value of x)
Answer:
Note: Typically transcribed as \(6^{2x+1} \div 6 = 216\) or similar standard problem. Let us solve assuming \(6^{2x+1} \div 6 = 6^3\):
\(6^{2x+1-1} = 6^3 \implies 6^{2x} = 6^3 \implies 2x = 3 \implies x = \frac{3}{2}\).
Alternatively, if equation is \(6^{2x+1} = 216 \times 6 = 1296 = 6^4\):
\(2x+1 = 4 \implies 2x = 3 \implies x = \frac{3}{2}\).
In simple words: Express both sides with base 6 and equate the exponents.

Exam Tip: Convert all constants into prime powers matching the base of the variable term.