CBSE Class 7 Mathematics Exponents and Powers Assignment Set 02

M. C. Q.

Question 1. If \(a^x = m\) then \(m\) is
(i) a power of \(a\)
(ii) an exponent of \(a\)
(iii) the base of \(x\)
(iv) the index of the power of \(a\)
Answer: (i) a power of \(a\)
In the exponential expression \(a^x = m\), \(a\) is the base, \(x\) is the exponent or index, and \(m\) is the result, which is referred to as the power of \(a\).
In simple words: The value obtained by raising a base to an exponent is called a power of that base.

Exam Tip: Remember the terminology: in \(a^x = m\), \(a\) = base, \(x\) = exponent/power, and \(m\) = value/power of \(a\).

Question 2. \(\left[\left(\frac{1}{9}\right)^{-\frac{1}{2}}\right]^3\) is equal to
(i) \(\frac{1}{27}\)
(ii) 27
(iii) 81
(iv) \(\frac{1}{81}\)
Answer: (ii) 27
Given expression = \(\left(\left(\frac{1}{9}\right)^{-\frac{1}{2}}\right)^3\)
First, simplify the inner term: \(\left(\frac{1}{9}\right)^{-\frac{1}{2}} = (9^{-1})^{-\frac{1}{2}} = 9^{\left(-1 \times -\frac{1}{2}\right)} = 9^{\frac{1}{2}} = \sqrt{9} = 3\)
Now, raise this result to the power of 3:
\(3^3 = 27\)
In simple words: First apply the negative square root to flip the fraction and get 3, then cube it to get 27.

Exam Tip: Use the property \((a^m)^n = a^{m \times n}\) to simplify nested exponents step by step.

Question 3. \(\left[\left\{\left(\frac{4}{-9}\right)^2\right\}^0\right]^{-\frac{1}{2}}\) equals
(i) zero
(ii) \(-\frac{4}{9}\)
(iii) 1
(iv) \(\frac{4}{9}\)
Answer: (iii) 1
Any non-zero expression raised to the power of 0 is equal to 1.
Inside the brackets: \(\left[\left\{\left(\frac{4}{-9}\right)^2\right\}^0\right]^{-\frac{1}{2}} = [1]^{-\frac{1}{2}} = 1\)
In simple words: Since any non-zero number to the power of zero is 1, the entire expression inside reduces to 1, and 1 to any power remains 1.

Exam Tip: Look out for a zero exponent in any part of a nested expression; it instantly simplifies that part to 1.

Question 4. The value of \((0.000729)^{\frac{5}{6}}\) is
(i) 0.243
(ii) 0.0243
(iii) 0.000243
(iv) 0.00243
Answer: (iv) 0.00243
Express 0.000729 as a fraction: \(0.000729 = \frac{729}{1000000} = \left(\frac{3}{10}\right)^6\)
Now apply the outer exponent \(\frac{5}{6}\):
\(\left(\left(\frac{3}{10}\right)^6\right)^{\frac{5}{6}} = \left(\frac{3}{10}\right)^{6 \times \frac{5}{6}} = \left(\frac{3}{10}\right)^5\)
\(= \frac{243}{100000} = 0.00243\)
In simple words: Convert the decimal to a fraction with a sixth power, multiply the powers, and convert back to a decimal.

Exam Tip: Count the decimal places carefully to determine the correct denominator (here 6 decimal places mean \(10^6\)).

Question 5. The value of \((0.000032)^{\frac{1}{5}}\) is
(i) 0.2
(ii) 2
(iii) 0.02
(iv) 0.002
Answer: (i) 0.2
Express 0.000032 as a fraction: \(0.000032 = \frac{32}{1000000} = \left(\frac{2}{10}\right)^5\)
Now apply the power \(\frac{1}{5}\):
\(\left(\left(\frac{2}{10}\right)^5\right)^{\frac{1}{5}} = \left(\frac{2}{10}\right)^{5 \times \frac{1}{5}} = \frac{2}{10} = 0.2\)
In simple words: Find the fifth root of 0.000032, which is 0.2 because \(0.2 \times 0.2 \times 0.2 \times 0.2 \times 0.2 = 0.000032\).

Exam Tip: Recognize standard powers like \(2^5 = 32\) and \(10^5 = 100000\).

Question 6. \((-3)^2 \times (-1)^{101}\) is equal to
(i) 9
(ii) -9
(iii) 6
(iv) -6
Answer: (ii) -9
Calculate each part separately:
\((-3)^2 = (-3) \times (-3) = 9\)
\((-1)^{101} = -1\) (since 101 is an odd number)
Multiply the results: \(9 \times (-1) = -9\)
In simple words: A negative number to an even power is positive, and to an odd power is negative. So \(9 \times (-1) = -9\).

Exam Tip: Remember: negative base with even exponent gives positive, while negative base with odd exponent gives negative.

Question 7. If \(x^3 = -64\) then the value of \(x\) is
(i) 64
(ii) 8
(iii) -4
(iv) 4
Answer: (iii) -4
Given \(x^3 = -64\)
Since \(-64 = (-4) \times (-4) \times (-4) = (-4)^3\),
Taking the cube root on both sides: \(x = -4\)
In simple words: Find a number which when multiplied by itself three times gives -64, which is -4.

Exam Tip: Cube roots of negative numbers are always negative.

Question 8. In \(\left(\frac{2}{3}\right)^4\), the base is
(i) 2
(ii) 3
(iii) 4
(iv) \(\frac{2}{3}\)
Answer: (iv) \(\frac{2}{3}\)
In an exponential expression \(\left(\text{Base}\right)^{\text{Exponent}}\), the number being multiplied repeatedly is the base. Here, the base is \(\frac{2}{3}\) and the exponent is 4.
In simple words: The entire fraction \(\frac{2}{3}\) is the base that gets multiplied 4 times.

Exam Tip: Do not confuse the numerator or denominator individually as the base when the whole fraction is enclosed in parentheses.

Question 9. The reciprocal of \(\left(-\frac{3}{7}\right)^2\) is
(i) \(\left(-\frac{7}{3}\right)^2\)
(ii) \(\left(\frac{3}{-7}\right)^2\)
(iii) \(\frac{9}{49}\)
(iv) \(\frac{49}{3}\)
Answer: (i) \(\left(-\frac{7}{3}\right)^2\)
First, evaluate the expression: \(\left(-\frac{3}{7}\right)^2 = \frac{9}{49}\)
The reciprocal of \(\frac{9}{49}\) is \(\frac{49}{9}\)
Examine option (i): \(\left(-\frac{7}{3}\right)^2 = \frac{49}{9}\), which matches the reciprocal.
In simple words: The reciprocal of a fraction raised to a power is obtained by reciprocating the base while keeping the exponent the same.

Exam Tip: The reciprocal of \((a/b)^n\) is \((b/a)^n\) or \((a/b)^{-n}\).

Question 10. \(\frac{4^0 + 5^0 + 6^0}{6^2 \div 6}\) is equal to
(i) \(\frac{1}{6}\)
(ii) \(\frac{1}{2}\)
(iii) \(\frac{1}{3}\)
(iv) \(\frac{1}{4}\)
Answer: (ii) \(\frac{1}{2}\)
Numerator: \(4^0 + 5^0 + 6^0 = 1 + 1 + 1 = 3\)
Denominator: \(6^2 \div 6 = 36 \div 6 = 6\)
Dividing numerator by denominator: \(\frac{3}{6} = \frac{1}{2}\)
In simple words: Simplify the numerator to 3 (since any number to power zero is 1) and denominator to 6, then reduce the fraction to \(\frac{1}{2}\).

Exam Tip: Evaluate numerators and denominators separately before dividing.