CBSE Class 8 Mathematics Exponents and Powers Assignments Set 10

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

Exam Tip: Remember that a negative exponent indicates the reciprocal of the base raised to the corresponding positive power.

Question 2. Simplify: \((-3)^2 \times \left(\frac{5}{3}\right)^2\)
Answer: Expand each term:
\((-3)^2 = 9\)
\(\left(\frac{5}{3}\right)^2 = \frac{5^2}{3^2} = \frac{25}{9}\)
Multiply the results:
\(9 \times \frac{25}{9} = 25\)
In simple words: Evaluate the squares separately and then multiply them together to find the final result.

Exam Tip: Notice that the \(9\) in the numerator and denominator cancel out directly, making the calculation very simple.

Question 3. Find the value of \(\left[\left(\frac{-2}{3}\right)^{-2}\right]^1\)
Answer: Using the power of a power law \((a^m)^n = a^{m \times n}\):
\(= \left(\frac{-2}{3}\right)^{-2 \times 1}\)
\(= \left(\frac{-2}{3}\right)^{-2}\)
Using the negative exponent rule \(\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n\):
\(= \left(\frac{3}{-2}\right)^2\)
\(= \frac{3^2}{(-2)^2} = \frac{9}{4}\)
In simple words: First multiply the powers, then flip the fraction inside to make the negative exponent positive, and square both parts.

Exam Tip: Flipping the fraction changes the sign of the exponent from negative to positive.

Question 4. Express \(3 \times 10^{-7}\) in usual form.
Answer: Since the exponent of 10 is negative 7, move the decimal point 7 places to the left:
\(3 \times 10^{-7} = 0.0000003\)
In simple words: A negative power of 10 means you shift the decimal point to the left by that many places.

Exam Tip: Count the total number of zeros carefully including the one before the decimal point.

Question 5. Evaluate \(6^{-2}\).
Answer: Using the negative exponent rule \(a^{-n} = \frac{1}{a^n}\):
\(6^{-2} = \frac{1}{6^2} = \frac{1}{36}\)
In simple words: Turn the number into a fraction with 1 on top and the base with a positive exponent on the bottom.

Exam Tip: Do not confuse negative exponents with negative numbers; \(6^{-2}\) 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 becomes 2.1 (which is between 1 and 10):
\(0.000021 = 2.1 \times 10^{-5}\)
In simple words: Standard form requires a number between 1 and 10 multiplied by a power of 10. Moving right gives a negative exponent.

Exam Tip: Count exactly how many places the decimal point moved to determine the exponent value.

Question 7. Write \(41860000000\) in standard form.
Answer: Shift the decimal point 10 places to the left to obtain 4.186:
\(41860000000 = 4.186 \times 10^{10}\)
In simple words: For large numbers, shift the decimal point to the left and use a positive exponent equal to the number of places moved.

Exam Tip: Retain all significant non-zero digits when writing numbers in standard form.

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

Exam Tip: Pad with zeros as necessary when shifting the decimal point beyond the given digits.

Question 9. Evaluate: \((2^{-1} + 3^{-1} + 4^{-1})^0\)
Answer: Any non-zero base raised to the power of 0 is equal to 1. Since the expression inside the parenthesis is not zero:
\((2^{-1} + 3^{-1} + 4^{-1})^0 = 1\)
In simple words: Anything raised to the power of zero equals 1, regardless of how complicated the inside expression is.

Exam Tip: Always check the exponent outside the bracket first before evaluating the inner terms.

Question 10. Find the value of \((4^{-1} \times 2^{-1}) \times 3^{-1}\).
Answer: Convert each term using the negative exponent rule:
\(4^{-1} = \frac{1}{4}, \quad 2^{-1} = \frac{1}{2}, \quad 3^{-1} = \frac{1}{3}\)
Multiply the 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 negative exponents as fractions and multiply all the numerators together and all the denominators together.

Exam Tip: Make sure to multiply all denominators correctly to get 24.

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: Separate the whole number and decimal parts, then multiply each digit by 10 raised to its place value position.

Exam Tip: Remember that digits to the left of the decimal have positive or zero exponents, while digits to the right have negative exponents.

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: Assign powers of 10 decreasing from 3 down to -2 for each digit from left to right.

Exam Tip: Include the term with zero even though its value is 0, to show complete place value expansion.

Question 13. Find \(x\) such that \((-2)^{x+1} \times (-2)^5 = (-2)^7\).
Answer: Using the product rule for exponents 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:
\(x + 6 = 7\)
\(x = 7 - 6 = 1\)
In simple words: Combine the exponents on the left by adding them, then set them equal to the exponent on the right because the bases are identical.

Exam Tip: When bases on both sides of an equation are equal, you can directly equate their exponents.

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

Exam Tip: Always prime-factorize the base first when asked to convert to a specific base like 2 or 3.

Question 15. Simplify \((-2)^{-3} \times (-3)^{-3} \times (4)^{-3}\) and write the answer in exponential form.
Answer: Since all terms have the same exponent \(-3\), use the rule \(a^m \times b^m = (ab)^m\):
\(= [(-2) \times (-3) \times 4]^{-3}\)
Multiply the numbers inside the brackets:
\(= [6 \times 4]^{-3}\)
\(= [24]^{-3}\)
In simple words: When exponents are identical, multiply the bases together and keep the common exponent.

Exam Tip: Pay close attention to negative signs when multiplying: \((-2) \times (-3) = +6\).

Question 16. Find the value of \((4^0 + 3^{-1}) \times 3^2\).
Answer: Evaluate each part inside the parenthesis:
\(4^0 = 1\)
\(3^{-1} = \frac{1}{3}\)
Substitute these values:
\(\left(1 + \frac{1}{3}\right) \times 3^2\)
\(= \left(\frac{4}{3}\right) \times 9\)
\(= \frac{4 \times 9}{3} = \frac{36}{3} = 12\)
In simple words: Simplify the terms inside the brackets first, find their sum, and then multiply by \(3^2\) which is 9.

Exam Tip: Remember that \(4^0 = 1\), not 0.

Question 17. Simplify: \(2^3 \times 2^{-5} \times 2^6\)
Answer: Using 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: Since all bases are 2, simply add all the exponents together.

Exam Tip: Combine positive numbers first before subtracting to avoid arithmetic errors.

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: Apply the negative exponent rule to invert each fraction:
\(= \left(\frac{2}{1}\right)^2 + \left(\frac{3}{1}\right)^2 + \left(\frac{4}{1}\right)^2\)
\(= 2^2 + 3^2 + 4^2\)
Calculate the squares:
\(= 4 + 9 + 16 = 29\)
In simple words: Invert each fraction to make the exponents positive, then square each number and add them up.

Exam Tip: Inverting a fraction raised to a negative power changes it to a whole number in these cases.

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

Exam Tip: Always apply the square to both top and bottom numbers.

Question 20. Simplify: \((-3)^4 \times \left(\frac{1}{3}\right)^4\)
Answer: Evaluate \((-3)^4 = 81\) and \(\left(\frac{1}{3}\right)^4 = \frac{1}{81}\):
\(= 81 \times \frac{1}{81}\)
\(= 1\)
Alternatively, use the property \(a^m \times b^m = (ab)^m\):
\(= \left[(-3) \times \frac{1}{3}\right]^4\)
\(= [-1]^4 = 1\)
In simple words: Multiply \(-3\) by \(\frac{1}{3}\) first to get \(-1\), then raise it to the power of 4.

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

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: Evaluate each term by inverting the fractions:
\(\left(\frac{1}{3}\right)^{-2} = 3^2 = 9\)
\(\left(\frac{1}{2}\right)^{-2} = 2^2 = 4\)
\(\left(\frac{1}{4}\right)^{-2} = 4^2 = 16\)
Substitute these values back into the expression:
\(= [9 - 4] \div 16\)
\(= 5 \div 16 = \frac{5}{16}\)
In simple words: Convert each negative exponent term into its positive integer value, subtract inside the brackets, and then divide by the last term.

Exam Tip: Follow the order of operations (PEMDAS/BODMAS): evaluate brackets first, then division.

Question 22. Evaluate \(\left(\frac{5}{9}\right)^{-7} \times \left(\frac{9}{5}\right)^{-4}\)
Answer: Rewrite the second term by inverting the fraction to make its exponent positive:
\(\left(\frac{9}{5}\right)^{-4} = \left(\frac{5}{9}\right)^4\)
Substitute this back into the expression:
\(= \left(\frac{5}{9}\right)^{-7} \times \left(\frac{5}{9}\right)^4\)
Using 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:
\(= \left(\frac{9}{5}\right)^3 = \frac{9^3}{5^3} = \frac{729}{125}\)
In simple words: Make the bases identical by flipping one fraction, add the exponents, and then simplify the final power.

Exam Tip: Making bases match is the key strategy for simplifying exponent multiplication.

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 the total thickness of the books:
Thickness of 6 books = \(6 \times 20 \text{ mm} = 120 \text{ mm}\)
Calculate the total thickness of the sheets:
Thickness of 5 sheets = \(5 \times 0.015 \text{ mm} = 0.075 \text{ mm}\)
Add both thicknesses together:
Total thickness = \(120 + 0.075 = 120.075 \text{ mm}\)
In simple words: Multiply the thickness of one book by the number of books, do the same for the sheets, and add both results together.

Exam Tip: Align decimal points properly when adding whole numbers and decimals.

Question 24. Find 'a' such that \((-5)^{a+2} \times (-5)^4 = (-5)^9\).
Answer: Use the product rule on the left side:
\((-5)^{(a+2)+4} = (-5)^9\)
\((-5)^{a+6} = (-5)^9\)
Since the bases are equal, equate the exponents:
\(a + 6 = 9\)
\(a = 9 - 6 = 3\)
In simple words: Add the exponents on the left side and set the sum equal to 9 because the bases are the same.

Exam Tip: Clearly show the step where you equate the exponents to secure full marks.

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 of the first term:
\(\left(\frac{3}{2}\right)^{-5} = \left(\frac{2}{3}\right)^5\)
Substitute back into the expression:
\(= \left(\frac{2}{3}\right)^{-7} \times \left(\frac{2}{3}\right)^5\)
Add the exponents using the product rule:
\(= \left(\frac{2}{3}\right)^{-7 + 5}\)
\(= \left(\frac{2}{3}\right)^{-2}\)
Invert the fraction 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 fraction, combine the exponents, and evaluate the final positive power.

Exam Tip: Reciprocating a fraction changes the sign of its exponent.

Question 26. Simplify \(\frac{25 \times a^{-4}}{5^{-3} \times 10 \times a^{-8}}\), Where \(a \neq 0\)
Answer: Express numbers in terms of prime factors (\(25 = 5^2\), \(10 = 2 \times 5\)):\;
\(= \frac{5^2 \times a^{-4}}{5^{-3} \times (2 \times 5^1) \times a^{-8}}\)
Combine powers of 5 in the denominator:
\(= \frac{5^2 \times a^{-4}}{2 \times 5^{-3+1} \times a^{-8}} = \frac{5^2 \times a^{-4}}{2 \times 5^{-2} \times a^{-8}}\)
Apply the division rule for exponents:\;
\(= \frac{1}{2} \times 5^{2 - (-2)} \times a^{-4 - (-8)}\)
\(= \frac{1}{2} \times 5^{2+2} \times a^{-4+8}\)
\(= \frac{1}{2} \times 5^4 \times a^4\)
\(= \frac{625}{2} a^4\)
In simple words: Break numbers into prime factors, then apply exponent laws by subtracting powers in the denominator from powers in the numerator.

Exam Tip: Be very careful with subtraction of negative exponents, such as \(-4 - (-8) = 4\).

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:
Red blood cell size = \(0.000007 \text{ m} = 7 \times 10^{-6} \text{ m}\)
Plant cell size = \(0.00001275 \text{ m} = 1.275 \times 10^{-5} \text{ m} = 12.75 \times 10^{-6} \text{ m}\)
To compare them, find the ratio:
\(\frac{\text{Size of red blood cell}}{\text{Size of plant cell}} = \frac{7 \times 10^{-6}}{1.275 \times 10^{-5}} = \frac{7 \times 10^{-6}}{12.75 \times 10^{-6}} = \frac{7}{12.75} \approx 0.55\)
In simple words: Convert both decimals to standard form with the same power of 10, then divide them to see how their sizes compare.

Exam Tip: Adjusting both numbers to have the same exponent of 10 makes division straightforward.

Question 28. Simplify \(\frac{3^{-5} \times 10^{-5} \times 125}{6^{-5} \times 5^{-7}}\)
Answer: Prime factorize composite numbers: \(10 = 2 \times 5\), \(125 = 5^3\), and \(6 = 2 \times 3\):
\(= \frac{3^{-5} \times (2 \times 5)^{-5} \times 5^3}{(2 \times 3)^{-5} \times 5^{-7}}\)
Expand the products raised to powers:
\(= \frac{3^{-5} \times 2^{-5} \times 5^{-5} \times 5^3}{2^{-5} \times 3^{-5} \times 5^{-7}}\)
Cancel out common terms (\(3^{-5}\) and \(2^{-5}\) appear in both numerator and denominator):
\(= \frac{5^{-5+3}}{5^{-7}} = \frac{5^{-2}}{5^{-7}}\)
Apply the division rule:\;
\(= 5^{-2 - (-7)} = 5^{-2+7} = 5^5 = 3125\)
In simple words: Convert all numbers into prime bases (2, 3, 5), expand terms, cancel matching terms from top and bottom, and simplify the remaining powers of 5.

Exam Tip: Prime factorization is the most reliable tool for simplifying complex fractional expressions with exponents.

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: A positive exponent means repeated multiplication, while a negative exponent means repeated division (the reciprocal).

Exam Tip: Remember that negative exponents never yield negative numbers.

Question 2. Find the value of \(\left(\frac{1}{2}\right)^{-1}\).
Answer: Using the negative exponent rule for fractions:\;
\(\left(\frac{1}{2}\right)^{-1} = \left(\frac{2}{1}\right)^1 = 2\)
In simple words: Raising a fraction to the power of -1 simply flips the fraction upside down.

Exam Tip: Any fraction to the power of -1 is 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 terms with the same base.

Exam Tip: Do not multiply the bases together; keep the base 2 unchanged.

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

Exam Tip: Power of a power means multiplication of exponents, not addition.

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

Exam Tip: Ensure the final answer specifically uses a positive exponent as requested.

Question 6. Simplify the following exponents:
\((-2)^2 \times (-3)^2 \times (-5)^2\)
Answer: Evaluate each squared term:
\((-2)^2 = 4\)
\((-3)^2 = 9\)
\((-5)^2 = 25\)
Multiply the values together:
\(= 4 \times 9 \times 25\)
\(= 36 \times 25 = 900\)
Alternatively, use the exponent combination rule:\;
\(= [(-2) \times (-3) \times (-5)]^2 = [-30]^2 = 900\)
In simple words: Square each number individually and multiply, or multiply the bases first and then square.

Exam Tip: Even powers eliminate 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 by adding exponents:
\(2^{2(n-1) + 3n} = 2^{n+1}\)
Expand the exponent: \(2n - 2 + 3n = 5n - 2\):
\(2^{5n-2} = 2^{n+1}\)
Since the 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 left side exponent, set it equal to the right side exponent, and solve the linear equation for \(n\).

Exam Tip: Be careful with distribution when expanding brackets in exponents like \(2(n-1)\).

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

Exam Tip: Group \(p\) terms and constant terms carefully to avoid sign errors.

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

Exam Tip: Powers decrease from left to right, starting with positive integers and moving into negative integers after the decimal point.

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: Adjust the decimal point so that the leading number is between 1 and 10, then count the number of places shifted for the exponent.

Exam Tip: Right shifts give negative exponents; left shifts give positive exponents.

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 exponents and right for positive exponents.

Exam Tip: Double-check the number of zeros added during the decimal shift.