CBSE Class 8 Mathematics Exponents and Powers Assignments Set 02

Question. The value of 2²⁰⁰¹ + 2¹⁹⁹⁹ / 2²⁰⁰⁰ - 2¹⁹⁹⁸, is ____.
(A) 2
(B) 10/3
(C) 2¹⁰⁰⁰ + 1
(D) 10
Answer : B

Question. When simplified, (x –1 + y –1)–1 is equal to ____.
(A) x + y
(B) xy/x + y
(C) x y
(D) 1/x y
Answer : B

Question. The value of x + x (xx) at x = 2 is ____.
(A) 10
(B) 16
(C) 18
(D) 36
Answer : A

Question. If ax = by = c^z and b² = ac, then y equals
(A) 2xz/x + z
(B) xz/2(x − z)
(C) xz/2(z − x)
(D) 2xz/(x − z)
Answer : A

Question. Find x, if (5/11)^-3 x (5/11)⁵ = (5/11)^x.
(A) 3
(B) 4
(C) 8
(D) 2
Answer : D

Question. The size of a red blood cell is 0.000007 m and the size of a plant cell is 0.00001275 m.
Find the ratio of the size of red blood cell to that of plant cell.
(A) 13 : 56
(B) 28 : 51
(C) 31 : 39
(D) 22 : 31
Answer : B

Question. Pluto is 5913000000 km away from the Sun.
Express this distance in standard form.
(A) 5.913 × 10^–11
(B) 5.913 × 10^–9
(C) 5.913 × 10⁶
(D) 5.913 × 10⁹
Answer : D

Question. Solve for y, if

(A) 1/2
(B) −19/18
(C) 3/10
(D) 12/17
Answer : B

Question. Simplify :

(A) 1
(B) (a + b + c)³
(C) a² + b² + c²
(D) 0
Answer : A

Question. If (6x)⁶ = 62³ = then find the value of x.
(A) 1
(B) √6
(C) 3√6
(D) 6√6
Answer : C

Question. Find the value of (0.000064)^5/6 ÷ (0.00032)^6/5 .
(A) 0.2
(B) 0.4
(C) 5
(D) 2.5
Answer : C

Question. The value of (7^–1 – 8^–1)–1 – (3^–1 – 4^–1)^–1 is 
(A) 56
(B) 12
(C) 68
(D) 44
Answer : C

Question. When 5 × 10^–3 – 2000 × 10^–6 is expressed in standard form, we will get
(A) 3 × 10^–2
(B) 3 × 10^–3
(C) 3 × 10³
(D) 3 × 10²
Answer : B

Question. If 3^{(x – y)} = 27 and 3^{(x + y)} = 243, then x is equal to _______.
(A) 0
(B) 2
(C) 4
(D) 6
Answer : C

Question. Number of prime factors in (216)^3/5 x (2500)^2/5 x (300)^1/5 is _______.
(A) 6
(B) 9
(C) 8
(D) None of these
Answer : D

Question. Match the following.

(A) (P) → (iii); (Q) → (i); (R) → (iv); (S) → (ii)
(B) (P) → (iv); (Q) → (i); (R) → (ii); (S) → (iii)
(C) (P) → (ii); (Q) → (iii); (R) → (iv); (S) → (i)
(D) (P) → (iii); (Q) → (i); (R) → (ii); (S) → (iv)
Answer : A

Question. If x = (3/2)² x (2/3)^-4, find the value of x^–2.
(A) (4/3)¹² 
(B) (2/3)¹² 
(C) (3/2)¹² 
(D) (3/4)¹² 
Answer : B

Question. Fill in the blanks.
(i) If m² = 27^2/3 × 16^–3/2, then m = P 
(ii) If ab = 1, then 1/1+1a^-1 + 1/1 + b^-1 = Q 
(iii) If x = (8^2/3 · 32^–2/5), then x^–5 = R 
        P          Q           R
(A) 5/4         0         10/7
(B) 1            1         5/16
(C) 3/8         1         1
(D) 7/8         0         7/8
Answer : C

Question. If  10/3 × 3^x – 3^{x – 1} = 81, then the value of x is ____.
(A) 2
(B) 1
(C) 3
(D) 0
Answer : C

Question. Which of the following values are equal ?
P. 14     Q. 40
R. 04     S. 41
(A) P and Q
(B) Q and R
(C) P and R
(D) P and S
Answer : A

Question. The cell of bacteria doubles itself after every 1 hour. How many cells will there be after 8 hours?
(A) 200 times of the original
(B) 2¹⁰ times of the original
(C) 2⁸ times of the original
(D) 2⁶ times of the original
Answer : C

19. At the end of the 20^th century, the world’s population was approximately 6.125 × 10⁹.
Express this population in usual form.
(A) 6.125 × 10¹⁰
(B) 6125000000
(C) 6125 × 10⁵
(D) 61250000
Answer : B

Question. Weight of moon is (7.346 × 10²²) kg and weight of Earth is (5.9724 × 10²⁴) kg. What is the total weight of both in standard form?
(A) 6.04 × 10²⁴ kg
(B) 7.08 × 10²² kg
(C) 5.98 × 10²⁴ kg
(D) 6.44 × 10²⁴ kg
Answer : A

Question. Simplify :

(A) (2/3)^-4
(B) 32/3
(C) 243/16
(D) 243/32
Answer : D

Question. If a = (2^–2 – 2^–3), b = (2^–3 – 2^–4) and c =(2^–4 – 2^–2) then find :
(i) a³ + b³ + c³ (ii) 10 abc
           (i)                      (ii)
(A) 9/2048                 7/2048
(B) 3/1024                 5/2048
(C) –3/1024             –10/2048
(D) –9/2048             –15/1024
Answer : D

M.C.Q
Choose the correct option from the following four alternative answers.

Question 1. \((27)^{-\frac{4}{3}}\), its simplified form is
(i) \(\frac{1}{81}\)
(ii) \(81\)
(iii) \(\frac{1}{9}\)
(iv) \(\frac{1}{27}\)
Answer: (i) \(\frac{1}{81}\)
\((27)^{-\frac{4}{3}} = (3^3)^{-\frac{4}{3}}\)
\(= 3^{3 \times \left(-\frac{4}{3}\right)}\)
\(= 3^{-4} = \frac{1}{3^4} = \frac{1}{81}\)
In simple words: Write 27 as \(3^3\) and multiply the powers to get \(3^{-4}\), which becomes \(\frac{1}{81}\).

Exam Tip: Always express numbers in base form (prime factorization) before applying exponent laws like \((a^m)^n = a^{m \times n}\).

Question 2. \(\frac{a^m \times a^n}{a^p}\) can be rewritten as
(i) \(a^{m-p-n}\)
(ii) \(a^{m+n-p}\)
(iii) \(a^{p-m+n}\)
(iv) \(a^{p-m-n}\)
Answer: (ii) \(a^{m+n-p}\)
Using laws of exponents: \(a^m \times a^n = a^{m+n}\)
\(\frac{a^{m+n}}{a^p} = a^{(m+n) - p} = a^{m+n-p}\)
In simple words: Add exponents when multiplying terms with the same base, and subtract the exponent of the denominator when dividing.

Exam Tip: Remember the product rule \(a^m \times a^n = a^{m+n}\) and quotient rule \(\frac{a^m}{a^n} = a^{m-n}\).

Question 3. \(a^{\frac{m}{n}}\) can be written as
(i) \(\sqrt[n]{a}\)
(ii) \(\sqrt[m]{a^n}\)
(iii) \(\sqrt[n]{a^m}\)
(iv) \(a^{-n}\)
Answer: (iii) \(\sqrt[n]{a^m}\)
By definition of fractional exponents: \(a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m}\)
In simple words: The numerator of a fractional exponent acts as a power, and the denominator acts as a root.

Exam Tip: The denominator of the exponent always goes into the index of the radical root.

Question 4. If \(a^x = 1\), the value of \(x\) is
(i) \(1\)
(ii) \(-1\)
(iii) \(a\)
(iv) \(0\)
Answer: (iv) \(0\)
Any non-zero base raised to the power of zero is equal to 1, i.e., \(a^0 = 1\) (where \(a \neq 0\)).
In simple words: Any number (except 0) raised to the power zero gives 1.

Exam Tip: Do not confuse this with \(a^1 = a\). Zero exponent always yields 1.

Question 5. The value of \(\left(\frac{1}{216}\right)^{-\frac{2}{3}}\) is
(i) \(36\)
(ii) \(\frac{1}{36}\)
(iii) \(6\)
(iv) \(\frac{1}{6}\)
Answer: (i) \(36\)
\(\left(\frac{1}{216}\right)^{-\frac{2}{3}} = (216)^{\frac{2}{3}}\)
\(= (6^3)^{\frac{2}{3}}\)
\(= 6^{3 \times \frac{2}{3}} = 6^2 = 36\)
In simple words: Flip the fraction to make the negative power positive, write 216 as \(6^3\), and simplify.

Exam Tip: A negative exponent can be eliminated by taking the reciprocal of the base: \(\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m\).

Question 6. The value of \(2^{2^{2^2}}\) is
(i) \(2^8\)
(ii) \(2^{16}\)
(iii) \(2^{32}\)
(iv) \(4\)
Answer: (ii) \(2^{16}\)
Evaluate the tower of exponents from top to bottom:
Innermost exponent: \(2^2 = 4\)
Next level: \(2^4 = 16\)
Final expression: \(2^{16}\)
In simple words: Solve exponent towers from the top down. Here, \(2^2 = 4\), then \(2^4 = 16\), so the answer is \(2^{16}\).

Exam Tip: Do not multiply the base exponents directly when dealing with exponent towers; evaluate from top to bottom.

Question 7. Write \(\sqrt[8]{2^{-3}}\) in exponent form with positive power.
(i) \(2^{\frac{3}{8}}\)
(ii) \(\frac{8}{2^3}\)
(iii) \(\frac{1}{2^{\frac{3}{8}}}\)
(iv) \(\frac{1}{\frac{8}{2^3}}\)
Answer: (iii) \(\frac{1}{2^{\frac{3}{8}}}\)
\(\sqrt[8]{2^{-3}} = (2^{-3})^{\frac{1}{8}} = 2^{-\frac{3}{8}}\)
To make the power positive, take the reciprocal: \(\frac{1}{2^{\frac{3}{8}}}\)
In simple words: Convert the root to a fractional power \(-\frac{3}{8}\), then move it to the denominator to make the power positive.

Exam Tip: \(\sqrt[n]{a^m} = a^{\frac{m}{n}}\), and \(a^{-m} = \frac{1}{a^m}\).

Question 8. The value of \((2^3)^2\) is
(i) \(2^9\)
(ii) \(\frac{1}{2^9}\)
(iii) \(2^5\)
(iv) \(2^6\)
Answer: (iv) \(2^6\)
Using the power of a power rule \((a^m)^n = a^{m \times n}\):
\((2^3)^2 = 2^{3 \times 2} = 2^6\)
In simple words: Multiply the powers together when an exponent is raised to another exponent.

Exam Tip: \((a^m)^n\) means multiply powers (\(m \times n\)), whereas \(a^m \times a^n\) means add powers (\(m + n\)).

Question 9. The value of \((1^2 + 2^2 + 3^2)^2\) is
(i) \(14^2\)
(ii) \(6^4\)
(iii) \(6^2\)
(iv) \(14\) क्विंटल (or 14)
Answer: (i) \(14^2\)
Evaluate inside the parentheses:
\(1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14\)
Now apply the outer square: \((14)^2 = 14^2\)
In simple words: First calculate the sum inside the bracket (\(1+4+9 = 14\)), then square the result.

Exam Tip: Always simplify expressions inside brackets completely before applying outer exponents.

Question 10. The value of \(x^{a-b} \times x^{b-c} \times x^{c-a}\) is
(i) \(x\)
(ii) \(1\)
(iii) \(x^{2a+2b+2c}\)
(iv) \(x^{a+b+c}\)
Answer: (ii) \(1\)
Since the bases are the same, add all the exponents:
\(x^{(a-b) + (b-c) + (c-a)}\)
\(= x^{a - b + b - c + c - a}\)
\(= x^0 = 1\)
In simple words: When multiplying identical bases, add all exponents together. All terms cancel out to 0, giving \(x^0 = 1\).

Exam Tip: Cyclic algebraic sums in exponents often cancel out to zero when multiplied.