CBSE Class 8 Mathematics Exponents and Powers Assignments Set 11

CHAPTER - 12

Exponents and Powers

Questions carrying 1 Mark each :-

Q.1 Write in exponential form. (Note: Assuming a standard textbook problem of repeated multiplication such as \(a \times a \times a \times a\), or taking a general expression). Let us assume the standard form for a repeated product like \(a \times a \times a \times a \times a\).
Answer: For a number multiplied by itself multiple times, such as \(a \times a \times a \times a \times a = a^5\).
In simple words: Count how many times the base number is multiplied by itself and write it as a power.

Exam Tip: The number of times a base is multiplied becomes the exponent.

Q.2 Find the multiplicative inverse of \(2^{-3}\).
Answer: The multiplicative inverse of a number is what you multiply it by to get 1.
Multiplicative inverse of \(2^{-3}\) is \(\frac{1}{2^{-3}} = 2^3\)
In simple words: To find the multiplicative inverse of a power, simply change the sign of the exponent.

Exam Tip: The multiplicative inverse of \(a^{-n}\) is \(a^n\).

Questions carrying 2 marks each:-

Q.3 Expand 253.45 using exponents.
Answer: Given number = 253.45
\(253.45 = (2 \times 10^2) + (5 \times 10^1) + (3 \times 10^0) + (4 \times 10^{-1}) + (5 \times 10^{-2})\)
In simple words: Multiply each digit by its respective place value expressed as a power of 10.

Exam Tip: Digits to the left of the decimal have positive powers of 10 starting from \(10^0\), and digits to the right have negative powers starting from \(10^{-1}\).

Q.4 Simplify and express the result as positive exponent :- \(\left(\frac{2}{3}\right)^{-3} \times \left(\frac{2}{3}\right)^{-2} \times \left(\frac{2}{3}\right)^{3}\)
Answer: Given expression = \(\left(\frac{2}{3}\right)^{-3} \times \left(\frac{2}{3}\right)^{-2} \times \left(\frac{2}{3}\right)^{3}\)
Using the laws of exponents \(a^m \times a^n = a^{m+n}\):
\(= \left(\frac{2}{3}\right)^{-3 + (-2) + 3}\)
\(= \left(\frac{2}{3}\right)^{-2}\)
Using the negative exponent rule \(a^{-m} = \left(\frac{1}{a}\right)^m\):
\(= \left(\frac{3}{2}\right)^2\)
In simple words: Add the exponents when bases are the same, then flip the fraction to make the final exponent positive.

Exam Tip: Always add exponents during multiplication with same bases before converting negative exponents to positive ones.

Q.5 Evaluate \(\left\{\left(\frac{-2}{3}\right)^{-2}\right\}^3\)
Answer: Given expression = \(\left\{\left(\frac{-2}{3}\right)^{-2}\right\}^3\)
Using the power of a power law \((a^m)^n = a^{m \times n}\):
\(= \left(\frac{-2}{3}\right)^{-2 \times 3}\)
\(= \left(\frac{-2}{3}\right)^{-6}\)
Using the negative exponent law to make the exponent positive:
\(= \left(\frac{3}{-2}\right)^6\)
\(= \frac{3^6}{(-2)^6}\)
\(= \frac{729}{64}\)
In simple words: Multiply the inner and outer exponents, invert the base to make the exponent positive, and then compute the powers.

Exam Tip: An even power of a negative number always yields a positive result.

Q.6 Write 0.00053 in standard form.
Answer: Shift the decimal point to the right until there is only one non-zero digit to its left.
\(0.00053 = 5.3 \times 10^{-4}\)
In simple words: Count how many places you move the decimal to the right and use that number as a negative exponent of 10.

Exam Tip: Standard form is written as \(k \times 10^n\) where \(1 \le |k| < 10\) and \(n\) is an integer.

Q.7 Express \(6.11 \times 10^{-6}\) in usual form.
Answer: Given number = \(6.11 \times 10^{-6}\)
Shift the decimal point 6 places to the left:
\(6.11 \times 10^{-6} = 0.00000611\)
In simple words: Move the decimal point to the left according to the negative power of 10.

Exam Tip: A negative exponent in scientific notation means the usual form is a small decimal number less than 1.

Questions carrying 3 marks each :-

Q.8 Solve for x : \(\left(\frac{-3}{3}\right)^{-3} \div \left(\frac{-3}{3}\right)^{-3} = \left(\frac{-3}{3}\right)^{2x}\) (Standard text equivalent: \((-3)^{-3} \div (-3)^{-3} = (-3)^{2x}\))
Answer: Given equation: \((-3)^{-3} \div (-3)^{-3} = (-3)^{2x}\)
Using the division law of exponents \(a^m \div a^n = a^{m-n}\):
\((-3)^{-3 - (-3)} = (-3)^{2x}\)
\((-3)^{-3 + 3} = (-3)^{2x}\)
\((-3)^0 = (-3)^{2x}\)
Since any non-zero number to the power 0 is 1, and equating exponents when bases are equal:
\(2x = 0\)
\(x = 0\)
In simple words: Simplify both sides using exponent rules, then equate the powers when the bases are the same.

Exam Tip: Remember that \(a^0 = 1\) for any \(a \neq 0\).

Q.9 Simplify : \(\left(\frac{3}{9}\right)^{-3} \times \left(\frac{6}{9}\right)^{-3} \div \left(\frac{-3}{9}\right)^{-3}\)
Answer: First, simplify the fractions inside the brackets:
\(\frac{3}{9} = \frac{1}{3}, \quad \frac{6}{9} = \frac{2}{3}, \quad \frac{-3}{9} = \frac{-1}{3}\)
Substitute these back into the expression:
\(= \left(\frac{1}{3}\right)^{-3} \times \left(\frac{2}{3}\right)^{-3} \div \left(\frac{-1}{3}\right)^{-3}\)
Using the property \(a^{-m} = \left(\frac{1}{a}\right)^m\):
\(= 3^3 \times \left(\frac{3}{2}\right)^3 \div (-3)^3\)
\(= 27 \times \frac{27}{8} \div (-27)\)
\(= 27 \times \frac{27}{8} \times \frac{1}{-27}\)
\(= \frac{27}{-8} = -\frac{27}{8}\)
In simple words: Reduce the fractions first, apply the negative exponent rule to invert them, and then perform multiplication and division.

Exam Tip: Simplifying fractions before applying powers makes calculations much easier.

Multiple choice Question Carrying 1 mark each :-

Q.10 \(a^0 = \_\_\_\_\_\_\), where \(a \neq 0\)
(a) 0
(b) a
(c) 1
(d) none of the options.
Answer: (c) 1
Any non-zero base raised to the power of 0 is always equal to 1.
In simple words: Anything to the power zero equals 1.

Exam Tip: This is a fundamental law of exponents: \(a^0 = 1\) for \(a \neq 0\).

Q.11 The reciprocal of \(\left(\frac{-b}{-a}\right)^n\) is
(a) \(\left(\frac{b}{-b}\right)^n\)
(b) \(\left(\frac{-b}{-a}\right)^{-n}\)
(c) \(\left(\frac{b}{b}\right)^n\)
(d) \((b)^n\)
Answer: (b) \(\left(\frac{-b}{-a}\right)^{-n}\)
The reciprocal of a term \(x^n\) is \(x^{-n}\). Therefore, the reciprocal of \(\left(\frac{-b}{-a}\right)^n\) is \(\left(\frac{-b}{-a}\right)^{-n}\).
In simple words: The reciprocal is obtained by changing the sign of the exponent.

Exam Tip: Reciprocal of \(a^n\) is always \(a^{-n}\).