CBSE Class 8 Mathematics Exponents And Powers Worksheet Set 04

Question 1. \((-2)^5 \div (-2)^8\)
Answer: Using the law of exponents for division, \(a^m \div a^n = a^{m-n}\):
\(= (-2)^{5-8}\)
\(= (-2)^{-3}\)
Using the negative exponent rule \(a^{-m} = \frac{1}{a^m}\):
\(= \frac{1}{(-2)^3} = -\frac{1}{8}\)
In simple words: When dividing identical bases, subtract the exponent of the denominator from the numerator.

Exam Tip: Remember that an odd negative power retains its negative sign: \((-2)^3 = -8\).

Question 2. Simplify: \((-3)^2 \times \left(\frac{5}{3}\right)^2\)
Answer: Expand each term:
\(= (-3) \times (-3) \times \frac{5^2}{3^2}\)
\(= 9 \times \frac{25}{9}\)
Cancel out the common factor of 9:
\(= 25\)
In simple words: Compute each square first and then multiply the resulting fractions and integers.

Exam Tip: Squaring a negative number always yields a positive result, so \((-3)^2 = 9\).

Question 3. Find the value of \(\left[\left(-\frac{2}{3}\right)^{-2}\right]^{-1}\)
Answer: Using the power of a power property \((a^m)^n = a^{m \times n}\):
\(= \left(-\frac{2}{3}\right)^{(-2) \times (-1)}\)
\(= \left(-\frac{2}{3}\right)^2\)
\(= \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) = \frac{4}{9}\)
In simple words: Multiply the exponents together first, then evaluate the resulting power.

Exam Tip: A negative number raised to an even power becomes positive.

Question 4. Express \(3 \times 10^{-7}\) in usual form.
Answer: Move the decimal point 7 places to the left since the exponent on 10 is \(-7\):
\(= 0.0000003\)
In simple words: Shift the decimal point to the left by the number of zeros indicated by the negative exponent.

Question 5. Evaluate \(6^{-2}\).
Answer: Using the negative exponent rule \(a^{-m} = \frac{1}{a^m}\):
\(= \frac{1}{6^2}\)
\(= \frac{1}{36}\)
In simple words: Flip the base to the denominator and change the negative exponent to positive.

Exam Tip: Do not confuse negative exponents with negative numbers; the result is a positive fraction.

Question 6. Write \(0.000021\) in standard form.
Answer: Shift the decimal point 5 places to the right so that the number is between 1 and 10:
\(= 2.1 \times 10^{-5}\)
In simple words: Standard form requires a number between 1 and 10 multiplied by a power of 10.

Exam Tip: Moving the decimal to the right results in a negative exponent for the base 10.

Question 7. Write \(41860000000\) in standard form.
Answer: Shift the decimal point 10 places to the left to obtain a number between 1 and 10:
\(= 4.186 \times 10^{10}\)
In simple words: Count how many places you move the decimal to the left to make the number less than 10.

Question 8. Express \(4.5 \times 10^4\) in usual form.
Answer: Move the decimal point 4 places to the right since the exponent is \(4\):
\(= 4.5 \times 10000\)
\(= 45000\)
In simple words: Multiply by 10000 by shifting the decimal point four places to the right.

Exam Tip: Fill empty decimal places with zeros.

Question 9. Evaluate: \((2^{-1} + 3^{-1} + 4^{-1})^0\)
Answer: Any non-zero number raised to the power of 0 is equal to 1:
\(= 1\)
In simple words: Anything raised to the power of zero equals 1, so no inner calculation is needed.

Exam Tip: Always check the outer exponent first; a zero exponent makes the entire expression 1 immediately.

Question 10. Find the value of \((4^{-1} \times 2^{-1}) \times 3^{-1}\)
Answer: Convert negative exponents to fractions:
\(= \left(\frac{1}{4} \times \frac{1}{2}\right) \times \frac{1}{3}\)
\(= \frac{1}{8} \times \frac{1}{3}\)
\(= \frac{1}{24}\)
In simple words: Rewrite each term as its reciprocal fraction and multiply the numerators and denominators.

Question 11. Expand \(1526.26\) using exponents.
Answer: Write each digit with its corresponding place value power of 10:
\(= (1 \times 10^3) + (5 \times 10^2) + (2 \times 10^1) + (6 \times 10^0) + (2 \times 10^{-1}) + (6 \times 10^{-2})\)
In simple words: Break down the number into whole and decimal parts, multiplying each digit by its power of 10.

Exam Tip: Whole number places start from \(10^0\) at the units place going left, and decimal places start from \(10^{-1}\) going right.

Question 12. Expand \(1026.13\) using exponents.
Answer: Write each digit with its corresponding place value power of 10:
\(= (1 \times 10^3) + (0 \times 10^2) + (2 \times 10^1) + (6 \times 10^0) + (1 \times 10^{-1}) + (3 \times 10^{-2})\)
In simple words: Multiply every digit by its matching positive or negative power of 10 based on its position.

Exam Tip: Do not omit the zero term; include \(0 \times 10^2\) for completeness in expanded exponential form.

Question 13. Find \(x\) such that \((-2)^{x+1} \times (-2)^5 = (-2)^7\).
Answer: Use the product rule on the left side (\(a^m \times a^n = a^{m+n}\)):
\((-2)^{(x+1)+5} = (-2)^7\)
\((-2)^{x+6} = (-2)^7\)
Since the bases are equal, equate the exponents:
\(x + 6 = 7\)
\(x = 7 - 6\)
\(x = 1\)
In simple words: Combine exponents on the left side by adding them, then set the powers equal since bases are identical.

Exam Tip: When bases on both sides of an equation are identical, you can drop the bases and solve for the exponent.

Question 14. Express \(8^{-3}\) as power with the base 2.
Answer: Express 8 as \(2^3\):
\(8^{-3} = (2^3)^{-3}\)
Using the power of a power property \((a^m)^n = a^{m \times n}\):
\(= 2^{3 \times (-3)}\)
\(= 2^{-9}\)
In simple words: Substitute 8 with \(2^3\) and multiply the exponents together.

Question 15. Simplify \((-2)^{-3} \times (-3)^{-3} \times (4)^{-3}\) and write the answer in exponential form.
Answer: Since all bases have the same exponent \(-3\), use the rule \(a^m \times b^m \times c^m = (abc)^m\):
\(= [(-2) \times (-3) \times 4]^{-3}\)
\(= (24)^{-3}\)
In simple words: When exponents are identical, multiply the bases together and keep the common exponent.

Exam Tip: Check your sign multiplication carefully: \((-2) \times (-3) = 6\), then \(6 \times 4 = 24\).

Question 16. Find the value of \((4^0 + 3^{-1}) \times 3^2\).
Answer: Evaluate terms inside the parentheses (\(4^0 = 1\) and \(3^{-1} = \frac{1}{3}\)):
\(= \left(1 + \frac{1}{3}\right) \times 9\)
\(= \left(\frac{4}{3}\right) \times 9\)
\(= 4 \times 3 = 12\)
In simple words: Simplify the sum inside the brackets first, convert \(3^{-1}\) to a fraction, then multiply.

Exam Tip: Always perform operations inside parentheses before multiplying by outside terms.

Question 17. Simplify: \(2^3 \times 2^{-5} \times 2^6\)
Answer: Use the product rule for exponents \(a^m \times a^n \times a^p = a^{m+n+p}\):
\(= 2^{3 + (-5) + 6}\)
\(= 2^{3 - 5 + 6}\)
\(= 2^4 = 16\)
In simple words: Add all exponents together when multiplying terms with the same base.

Exam Tip: Be careful with signs when adding negative numbers in the exponent sum.

Question 18. Find the value of \(\left(\frac{1}{2}\right)^{-2} + \left(\frac{1}{3}\right)^{-2} + \left(\frac{1}{4}\right)^{-2}\)
Answer: Use the negative exponent rule for fractions \(\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m\):
\(= \left(\frac{2}{1}\right)^2 + \left(\frac{3}{1}\right)^2 + \left(\frac{4}{1}\right)^2\)
\(= 2^2 + 3^2 + 4^2\)
\(= 4 + 9 + 16 = 29\)
In simple words: Reciprocal each fraction to make the negative exponents positive, then calculate each square and add.

Exam Tip: Flipping a fraction turns a negative outer exponent into a positive one instantly.

Question 19. Find the value of \(\left(\frac{5}{7}\right)^{-2}\)
Answer: Reciprocal the fraction to make the exponent positive:
\(= \left(\frac{7}{5}\right)^2\)
\(= \frac{7^2}{5^2} = \frac{49}{25}\)
In simple words: Invert the fraction and square both numerator and denominator.

Exam Tip: \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\) is a major time-saving property.

Question 20. Simplify: \((-3)^4 \times \left(\frac{1}{3}\right)^4\)
Answer: Since \((-3)^4 = 3^4\):
\(= 3^4 \times \frac{1^4}{3^4}\)
\(= 3^4 \times \frac{1}{3^4}\)
\(= \frac{3^4}{3^4} = 1\)
In simple words: Convert \((-3)^4\) to positive \(3^4\) and cancel out with the denominator term.

Exam Tip: Even powers eliminate negative signs, so \((-3)^4 = 3^4\).

Question 21. Simplify \(\left[\left(\frac{1}{3}\right)^{-2} - \left(\frac{1}{2}\right)^{-2}\right] \div \left(\frac{1}{4}\right)^{-2}\)
Answer: Simplify each bracket by taking reciprocals:
\(= \left[\left(\frac{3}{1}\right)^2 - \left(\frac{2}{1}\right)^2\right] \div \left(\frac{4}{1}\right)^2\)
\(= [3^2 - 2^2] \div 4^2\)
\(= [9 - 4] \div 16\)
\(= 5 \div 16 = \frac{5}{16}\)
In simple words: Evaluate the bracket terms first by clearing negative exponents, subtract, then divide by the final term.

Question 22. Evaluate \(\left(\frac{5}{9}\right)^{-7} \times \left(\frac{9}{5}\right)^{-4}\)
Answer: Rewrite the second term by inverting the base to make its exponent positive:
\(= \left(\frac{5}{9}\right)^{-7} \times \left(\frac{5}{9}\right)^4\)
Now use the product rule for exponents (\(a^m \times a^n = a^{m+n}\)):
\(= \left(\frac{5}{9}\right)^{-7 + 4}\)
\(= \left(\frac{5}{9}\right)^{-3}\)
Invert the base to make the exponent positive:
\(= \left(\frac{9}{5}\right)^3 = \frac{729}{125}\)
In simple words: Make the bases identical by reciprocating one fraction, combine the powers, and then simplify.

Exam Tip: Making bases match is the key strategy in exponential simplification problems.

Question 23. In a stack there are 6 books each of thickness 20 mm and 5 sheets each of thickness 0.015 mm. Find the total thickness of the stack?
Answer: Calculate total thickness of books:
Thickness of 6 books = \(6 \times 20\text{ mm} = 120\text{ mm}\)
Calculate total thickness of sheets:
Thickness of 5 sheets = \(5 \times 0.015\text{ mm} = 0.075\text{ mm}\)
Total thickness of the stack = \(120 + 0.075 = 120.075\text{ mm}\)
In simple words: Multiply the count of each item by its individual thickness, then add both values together.

Exam Tip: Align decimal points carefully when adding the thickness of books and sheets.

Question 24. Find \(a\) such that \((-5)^{a+2} \times (-5)^4 = (-5)^9\).
Answer: Apply the product rule on the left side:
\((-5)^{(a+2)+4} = (-5)^9\)
\((-5)^{a+6} = (-5)^9\)
Since bases are equal, equate the exponents:
\(a + 6 = 9\)
\(a = 9 - 6\)
\(a = 3\)
In simple words: Add the exponents on the left side and equate the resulting expression to the right-hand exponent.

Exam Tip: Equating exponents is valid only when the bases on both sides are exactly identical.

Question 25. Simplify \(\left(\frac{2}{3}\right)^{-7} \times \left(\frac{3}{2}\right)^{-5}\)
Answer: Invert the second fraction to match the base \(\frac{2}{3}\):
\(= \left(\frac{2}{3}\right)^{-7} \times \left(\frac{2}{3}\right)^5\)
Combine exponents using the product rule:
\(= \left(\frac{2}{3}\right)^{-7 + 5}\)
\(= \left(\frac{2}{3}\right)^{-2}\)
Invert the base to make the exponent positive:
\(= \left(\frac{3}{2}\right)^2 = \frac{9}{4}\)
In simple words: Make the bases the same by reciprocating one term, add the powers, and invert to clear the negative sign.

Exam Tip: \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\) helps synchronize bases quickly.

Question 26. Simplify \(\frac{25 \times a^{-4}}{5^{-3} \times 10 \times a^{-8}}\), Where \(a \neq 0\)
Answer: Rewrite numbers in prime factor form (\(25 = 5^2\) and \(10 = 5 \times 2\)):
\(= \frac{5^2 \times a^{-4}}{5^{-3} \times (5 \times 2) \times a^{-8}}\)
\(= \frac{5^2 \times a^{-4}}{5^{-2} \times 2 \times a^{-8}}\)
Apply exponent rules for division (\(a^m \div a^n = a^{m-n}\)):
\(= \frac{5^{2 - (-2)} \times a^{-4 - (-8)}}{2}\)
\(= \frac{5^{2+2} \times a^{-4+8}}{2}\)
\(= \frac{5^4 \times a^4}{2} = \frac{625a^4}{2}\)
In simple words: Factor numbers into primes, combine powers of like bases, and subtract denominator exponents from numerator exponents.

Exam Tip: Be extra careful with subtraction of negative exponents: \(-4 - (-8) = -4 + 8 = 4\).

Question 27. Compare the size of a red blood cell which is \(0.000007\text{ m}\) to that of a plant cell which is \(0.00001275\text{ m}\).
Answer: Express both sizes in standard form:
Size of red blood cell = \(7 \times 10^{-6}\text{ m}\)
Size of plant cell = \(1.275 \times 10^{-5}\text{ m} = 12.75 \times 10^{-6}\text{ m}\)
To compare, find the ratio of the plant cell size to the red blood cell size:
Ratio = \(\frac{1.275 \times 10^{-5}}{7 \times 10^{-6}} = \frac{1.275 \times 10}{7} = \frac{12.75}{7} \approx 1.82\)
In simple words: Convert both decimals to standard form with the same power of 10, then divide to find how many times larger one cell is than the other.

Exam Tip: Standard form makes comparing very large or very small numbers straightforward.

Question 28. Simplify \(\frac{3^{-5} \times 10^{-5} \times 125}{6^{-5} \times 5^{-7}}\)
Answer: Express composite numbers in prime factorization (\(10 = 2 \times 5\), \(125 = 5^3\), and \(6 = 2 \times 3\)):
\(= \frac{3^{-5} \times (2 \times 5)^{-5} \times 5^3}{(2 \times 3)^{-5} \times 5^{-7}}\)
\(= \frac{3^{-5} \times 2^{-5} \times 5^{-5} \times 5^3}{2^{-5} \times 3^{-5} \times 5^{-7}}\)
Cancel out identical terms (\(3^{-5}\) and \(2^{-5}\)) from numerator and denominator:
\(= \frac{5^{-5+3}}{5^{-7}} = \frac{5^{-2}}{5^{-7}}\)
Apply the division rule for exponents:
\(= 5^{-2 - (-7)} = 5^{-2+7} = 5^5 = 3125\)
In simple words: Break all numbers into prime factors, apply exponent distribution, cancel common terms, and simplify.

Exam Tip: Prime factorization is the most reliable tool for simplifying complex exponential fractions.

Most Important Questions

Question 1. Find the value of
a) \(2^2\)
b) \(2^{-2}\)
Answer:
a) \(2^2 = 2 \times 2 = 4\)
b) \(2^{-2} = \frac{1}{2^2} = \frac{1}{4}\)
In simple words: Evaluate the positive power directly, and use the reciprocal rule for the negative power.

Exam Tip: A negative exponent does not make the number negative; it creates a fraction.

Question 2. Find the value of \((1/2)^{-1}\).
Answer: Using the reciprocal rule for fractions with negative exponents:\(\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}\):
\(= \left(\frac{2}{1}\right)^1 = 2\)
In simple words: Inverting a fraction flips its numerator and denominator.

Exam Tip: Raising any fraction to the power of \(-1\) simply gives its reciprocal.

Question 3. Calculate the value of \(2^3 \times 2^4\)
Answer: Using the product rule \(a^m \times a^n = a^{m+n}\):
\(= 2^{3+4}\)
\(= 2^7 = 128\)
In simple words: Add the exponents when multiplying identical bases.

Question 4. What is the exponential notation of \((2^3)^5\)
Answer: Using the power of a power property \((a^m)^n = a^{m \times n}\):
\(= 2^{3 \times 5}\)
\(= 2^{15}\)
In simple words: Multiply the inner exponent by the outer exponent.

Exam Tip: Do not confuse power of a power with the product rule; powers are multiplied, not added here.

Question 5. Simplify and express in positive exponential form.
\((-3)^{-7} \times (-3)^{-8}\)
Answer: Use the product rule for exponents:
\(= (-3)^{-7 + (-8)}\)
\(= (-3)^{-15}\)
Convert to positive exponential form:
\(= \frac{1}{(-3)^{15}}\)
In simple words: Add the negative exponents together, then place the result in the denominator with a positive exponent.

Exam Tip: Ensure your final answer explicitly shows a positive exponent as requested.

Question 6. Simplify the following exponents
\((-2)^2 \times (-3)^2 \times (-5)^2\)
Answer: Evaluate each squared term:
\(= 4 \times 9 \times 25\)
Multiply the numbers:
\(= 36 \times 25 = 900\)
In simple words: Calculate each square individually, noting that all negative signs become positive, then multiply.

Exam Tip: Squaring negative numbers eliminates all negative signs.

Question 7. Find the value of \(n\) in the given expression:
\(2^{2(n-1)} \times 2^{3n} = 2^{n+1}\)
Answer: Apply the product rule on the left side:
\(2^{2n - 2 + 3n} = 2^{n+1}\)
\(2^{5n - 2} = 2^{n+1}\)
Since bases are equal, equate the exponents:
\(5n - 2 = n + 1\)
\(5n - n = 1 + 2\)
\(4n = 3\)
\(n = \frac{3}{4}\)
In simple words: Simplify the exponent on the left by distributing and combining terms, then equate both exponents.

Exam Tip: Always distribute numbers outside brackets in exponents carefully before combining like terms.

Question 8. Find the value of \(p\) in the given expression
\((-3)^{p-3} \times (-3)^{-3p + 1} = (-3)^5\)
Answer: Combine exponents on the left side:
\((-3)^{(p - 3) + (-3p + 1)} = (-3)^5\)
\((-3)^{p - 3 - 3p + 1} = (-3)^5\)
\((-3)^{-2p - 2} = (-3)^5\)
Equate the exponents since bases are equal:
\(-2p - 2 = 5\)
\(-2p = 5 + 2\)
\(-2p = 7\)
\(p = -\frac{7}{2}\)
In simple words: Add the algebraic expressions in the exponents, simplify, and solve the linear equation for \(p\).

Exam Tip: Group like terms (\(p\) terms together and constant terms together) when solving for the variable.

Question 9. Write the given expressions in expanded form.
a) \(3456.234\)
b) \(2016.123\)
Answer:
a) \(3456.234 = (3 \times 10^3) + (4 \times 10^2) + (5 \times 10^1) + (6 \times 10^0) + (2 \times 10^{-1}) + (3 \times 10^{-2}) + (4 \times 10^{-3})\)
b) \(2016.123 = (2 \times 10^3) + (0 \times 10^2) + (1 \times 10^1) + (6 \times 10^0) + (1 \times 10^{-1}) + (2 \times 10^{-2}) + (3 \times 10^{-3})\)
In simple words: Multiply each digit by its respective place value power of 10.

Exam Tip: Remember that decimal places correspond to negative powers of 10 starting from \(-1\).

Question 10. Express the following numbers in their standard form.
a) \(0.0000000000468\)
b) \(0.345\)
c) \(372400000000000\)
d) \(76100000000\)
Answer:
a) \(4.68 \times 10^{-11}\)
b) \(3.45 \times 10^{-1}\)
c) \(3.724 \times 10^{14}\)
d) \(7.61 \times 10^{10}\)
In simple words: Shift the decimal so that only one non-zero digit appears to its left, and adjust the power of 10 accordingly.

Exam Tip: Moving decimal to the right gives a negative exponent; moving to the left gives a positive exponent.

Question 11. Express the following numbers in their usual form:
a) \(3.05 \times 10^{-6}\)
b) \(4.5 \times 10^4\)
c) \(3.1563 \times 10^6\)
Answer:
a) \(0.00000305\)
b) \(45000\)
c) \(3156300\)
In simple words: Shift the decimal point left for negative powers of 10 and right for positive powers of 10.

Exam Tip: Count the shifts carefully and pad with zeros as needed.