CBSE Class 7 Mathematics Exponents And Powers Worksheet Set 04

Question. Find the base and exponent of 7⁶.
Answer : Base = 7 Exponent = 6

Question. Find the base and exponent of (–41)⁹¹.
Answer : Base = – 41 Exponent = 91

Question. What is the square of 21?
Answer : 441

Question. 7³ × 7⁷ = _________
Answer : 7¹⁰

Question. 4¹⁴ × 4⁴ = _________
Answer : 4¹⁸

Question. 4⁴⁸ ÷ 4⁶ = _________
Answer : 4⁴²

Question. Give exponential form for (–P) × (–P) × (–P) × (–P) × (–P)
Answer : (– P)⁵

Question. Give exponential form for 8324 × 8324 × 8324 × 8324
Answer : (8324)⁴

Question. Find the value of (–1)⁵⁰¹?
Answer : –1

Question. 3³ × 27² = 3⁰
Answer : 3⁹

Question. 25⁵ × 125⁴ = 5⁰
Answer : 5²²

Question. 100⁷ ÷ 10⁴ = 10⁰
Answer : (10)¹

Question. 5⁽²⁵⁾ = 5^x find x.
Answer : 32

Question. What is the cube of 11?
Answer : 1331

Question. What is the square of –16?
Answer : 256

Question. What is the cube of –50?
Answer : 125000

Question. (64)³ = (4)^x find x.
Answer : x = 9

Question. Find the value of x if (2⁶ 2^–3) × 2¹⁴ = 2^x
Answer : 23

Question 1. Find the value of each of the following
a. \( 13^2 \)
b. \( 5^3 \)
c. \( 2^4 \)
d. \( 11^2 \)
e. \( (-3)^3 \)
f. \( (-1)^6 \)
Answer:
To find the value of these exponents, multiply the base number by itself as many times as the power shows:
a. \( 13^2 = 13 \times 13 = 169 \)
b. \( 5^3 = 5 \times 5 \times 5 = 125 \)
c. \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)
d. \( 11^2 = 11 \times 11 = 121 \)
e. \( (-3)^3 = (-3) \times (-3) \times (-3) = -27 \)
f. \( (-1)^6 = (-1) \times (-1) \times (-1) \times (-1) \times (-1) \times (-1) = 1 \)
In simple words: Multiplying a number by itself tells us its power value. Remember that negative numbers raised to odd powers stay negative, while raised to even powers become positive.

Exam Tip: Be careful with signs. A negative base like \( (-3) \) with an odd power always gives a negative answer, while \( (-1) \) with an even power gives a positive answer.

Question 2. Simplify
a. \( 3 \times 10^2 \)
b. \( 2^2 \times 5^3 \)
c. \( 0 \times 10^4 \)
d. \( \left(\frac{3}{4}\right)^3 \)
e. \( \left(\frac{-2}{3}\right)^4 \)
Answer:
Work out the value of each part first, then multiply:
a. \( 3 \times 10^2 = 3 \times 100 = 300 \)
b. \( 2^2 \times 5^3 = 4 \times 125 = 500 \)
c. \( 0 \times 10^4 = 0 \times 10000 = 0 \) (multiplying any number by zero results in zero)
d. \( \left(\frac{3}{4}\right)^3 = \frac{3 \times 3 \times 3}{4 \times 4 \times 4} = \frac{27}{64} \)
e. \( \left(\frac{-2}{3}\right)^4 = \frac{(-2) \times (-2) \times (-2) \times (-2)}{3 \times 3 \times 3 \times 3} = \frac{16}{81} \)
In simple words: Calculate the exponents first, then multiply the results. Remember that any number multiplied by zero is always zero.

Exam Tip: For fractions raised to a power, raise both the top and bottom numbers to that power separately to find the final value.

Question 3. Express each of the following in exponential form
a. \( \left(\frac{-5}{7}\right) \times \left(\frac{-5}{7}\right) \times \left(\frac{-5}{7}\right) \times \left(\frac{-5}{7}\right) \)
b. \( -5 \times -5 \times -5 \)
c. \( x \times x \times x \times x \times a \times a \times b \times b \times b \)
d. \( (-2) \times (-2) \times (-2) \times (-2) \times a \times a \times a \)
Answer:
To write these in exponential form, count how many times each number or variable is multiplied by itself and use that count as the power:
a. Since \( \left(\frac{-5}{7}\right) \) is multiplied 4 times:
\( \left(\frac{-5}{7}\right)^4 \)

b. Since \( -5 \) is multiplied 3 times:
\( (-5)^3 \)

c. Group the same letters together and count them:
\( x \) is multiplied 4 times, \( a \) is multiplied 2 times, and \( b \) is multiplied 3 times:
\( x^4 \cdot a^2 \cdot b^3 \) (or \( x^4 a^2 b^3 \))

d. Count the numbers and letters:
\( -2 \) is multiplied 4 times, and \( a \) is multiplied 3 times:
\( (-2)^4 \cdot a^3 \) (or \( (-2)^4 a^3 \))
In simple words: Count how many times the same number or letter is written. That count becomes the little power number at the top.

Exam Tip: Always use brackets around negative bases (like \( (-5)^3 \) or \( (-2)^4 \)) to show that the negative sign is also raised to that power.

Question 4. Express each of the following numbers as a product of powers of their prime factors.
a. 36
b. 675
c. 392
d. 864
e. 450
f. 1800
Answer:
Find the prime factors of each number by dividing them step-by-step:
a. \( 36 = 2 \times 18 = 2 \times 2 \times 9 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2 \)
b. \( 675 = 3 \times 225 = 3 \times 3 \times 75 = 3 \times 3 \times 3 \times 25 = 3 \times 3 \times 3 \times 5 \times 5 = 3^3 \times 5^2 \)
c. \( 392 = 2 \times 196 = 2 \times 2 \times 98 = 2 \times 2 \times 2 \times 49 = 2 \times 2 \times 2 \times 7 \times 7 = 2^3 \times 7^2 \)
d. \( 864 = 2 \times 432 = 2 \times 2 \times 216 = 2^3 \times 108 = 2^4 \times 54 = 2^5 \times 27 = 2^5 \times 3^3 \)
e. \( 450 = 2 \times 225 = 2 \times 3 \times 75 = 2 \times 3 \times 3 \times 25 = 2 \times 3^2 \times 5^2 \)
f. \( 1800 = 2 \times 900 = 2 \times 2 \times 450 = 2^3 \times 225 = 2^3 \times 3^2 \times 5^2 \)
In simple words: Break down each number into prime numbers like 2, 3, 5, and 7, then write them as powers.

Exam Tip: Use prime factorization trees or the continuous division method to ensure you do not miss any prime factors.

Question 5. Using laws of exponents , simplify
(i) \( 3^6 \times 3^5 \)
(ii) \( (7^2)^3 \div 7^3 \)
(iii) \( 2^{20} \div 2^5 \)
(iv) \( 2^4 \times 5^4 \)
(v) \( (2^0 + 3^0)(4^0 + 6^0) \)
(vi) \( \frac{7^3}{5^3} \)
Answer:
Use standard laws of exponents to find the answers:
(i) Since the bases are the same, add the powers when multiplying:
\( 3^6 \times 3^5 = 3^{6+5} = 3^{11} \)

(ii) Multiply powers for power-of-a-power, then subtract when dividing:
\( (7^2)^3 \div 7^3 = 7^{2 \times 3} \div 7^3 = 7^6 \div 7^3 = 7^{6-3} = 7^3 \)

(iii) Subtract the powers when dividing with the same base:
\( 2^{20} \div 2^5 = 2^{20-5} = 2^{15} \)

(iv) Multiply the bases together when the powers are the same:
\( 2^4 \times 5^4 = (2 \times 5)^4 = 10^4 \)

(v) Remember that any non-zero number raised to the power of 0 is 1:
\( (2^0 + 3^0)(4^0 + 6^0) = (1 + 1)(1 + 1) = 2 \times 2 = 4 \)

(vi) Combine the division under a single power:
\( \frac{7^3}{5^3} = \left(\frac{7}{5}\right)^3 \)
In simple words: When bases are the same, add powers to multiply and subtract powers to divide. Any number with a zero power is always 1.

Exam Tip: Clearly write down the exponent rule you are using (e.g., \( a^m \times a^n = a^{m+n} \)) in the margins to earn step-wise marks.

Question 6. Simplify and express each of the following in exponential form :
(i) \( \frac{2^{15}}{2^7 \times 2^3} \)
(ii) \( (3^5 \times 3^2)^3 \)
(iii) \( [(2^3)^4 \times 2^8] \div 2^{12} \)
(iv) \( \frac{5^4 \times x^{10} y^5}{5^4 \times x^7 y^4} \)
(v) \( \left(\frac{2}{3}\right)^5 \times \left(\frac{3}{5}\right)^5 \)
(vi) \( \frac{9^8 \times (x^2)^5}{(27)^4 \times (x^3)^2} \)
(vii) \( \frac{3^2 \times 7^8 \times 13^6}{21^2 \times 91^3} \)
(viii) \( \frac{10 \times 5^{n+1} + 25 \times 5^n}{3 \times 5^{n+2} + 10 \times 5^{n+1}} \)
Answer:
Let us solve each expression step-by-step:
(i) Add the powers in the denominator, then subtract from the numerator:
\( \frac{2^{15}}{2^7 \times 2^3} = \frac{2^{15}}{2^{10}} = 2^{15-10} = 2^5 \)

(ii) Add the powers inside, then multiply by the outer power:
\( (3^5 \times 3^2)^3 = (3^{5+2})^3 = (3^7)^3 = 3^{7 \times 3} = 3^{21} \)

(iii) Work out the brackets first, then divide:
\( [(2^{12}) \times 2^8] \div 2^{12} = [2^{20}] \div 2^{12} = 2^{20-12} = 2^8 \)

(iv) Divide matching bases by subtracting their powers:
\( \frac{5^4}{5^4} \times \frac{x^{10}}{x^7} \times \frac{y^5}{y^4} = 1 \times x^{10-7} \times y^{5-4} = x^3 y \)

(v) Combine bases under the same power:
\( \left( \frac{2}{3} \times \frac{3}{5} \right)^5 = \left( \frac{2}{5} \right)^5 \)

(vi) Convert numbers to base 3:
\( \frac{(3^2)^8 \times x^{10}}{(3^3)^4 \times (x^6)} = \frac{3^{16} \times x^{10}}{3^{12} \times x^6} = 3^4 x^4 = (3x)^4 \)

(vii) Convert composite bases into prime bases:
\( \frac{3^2 \times 7^8 \times 13^6}{(3 \times 7)^2 \times (7 \times 13)^3} = \frac{3^2 \times 7^8 \times 13^6}{3^2 \times 7^2 \times 7^3 \times 13^3} = \frac{3^2 \times 7^8 \times 13^6}{3^2 \times 7^5 \times 13^3} = 7^{8-5} \times 13^{6-3} = 7^3 \times 13^3 \) (which can also be written as \( 91^3 \))

(viii) Pull out \( 5^n \) as a common factor in both numerator and denominator:
\( \text{Numerator: } 10 \times 5^n \times 5^1 + 25 \times 5^n = 5^n(50 + 25) = 75 \times 5^n \)
\( \text{Denominator: } 3 \times 5^n \times 5^2 + 10 \times 5^n \times 5^1 = 5^n(75 + 50) = 125 \times 5^n \)
\( \frac{75 \times 5^n}{125 \times 5^n} = \frac{75}{125} = \frac{3}{5} \)
In simple words: Group matching bases together and apply the rules of exponents step-by-step. Keep work neat to avoid missing any terms.

Exam Tip: When dealing with variables (like \( x \)), remember that \( x \) is the same as \( x^1 \). This helps when adding or subtracting exponents.

Question 7. Write the numbers in expanded forms :
a) 20068
(b) 423719
(c) 680071
(d) 5004132
Answer:
Write each number using powers of 10 based on its place value:
a) \( 20068 = 2 \times 10^4 + 0 \times 10^3 + 0 \times 10^2 + 6 \times 10^1 + 8 \times 10^0 \)

b) \( 423719 = 4 \times 10^5 + 2 \times 10^4 + 3 \times 10^3 + 7 \times 10^2 + 1 \times 10^1 + 9 \times 10^0 \)

c) \( 680071 = 6 \times 10^5 + 8 \times 10^4 + 0 \times 10^3 + 0 \times 10^2 + 7 \times 10^1 + 1 \times 10^0 \)

d) \( 5004132 = 5 \times 10^6 + 0 \times 10^5 + 0 \times 10^4 + 4 \times 10^3 + 1 \times 10^2 + 3 \times 10^1 + 2 \times 10^0 \)
In simple words: Write out the value of each digit by multiplying it by 10 raised to the correct power for its position.

Exam Tip: Do not skip writing the zero terms (like \( 0 \times 10^3 \)) unless specified, as showing them keeps your place values perfectly aligned.

Question 8. Find the number :
(a) \( 5 \times 10^5 + 4 \times 10^4 + 2 \times 10^3 + 3 \times 10^0 \)
(b) \( 9 \times 10^6 + 8 \times 10^4 + 7 \times 10^2 + 6 \times 10^0 \)
(c) \( 3 \times 10^4 + 4 \times 10^3 + 5 \times 10^0 \)
Answer:
Convert the powers of 10 back to standard numbers and add them up:
(a) \( 500000 + 40000 + 2000 + 3 = 542003 \)

(b) \( 9000000 + 80000 + 700 + 6 = 9080706 \)

(c) \( 30000 + 4000 + 5 = 34005 \)
In simple words: Put the digits into their correct place values. Write 0 in any empty slots where a power of 10 is missing.

Exam Tip: Be very careful when a power is missing (like \( 10^1 \) or \( 10^2 \)). Remember to put a 0 in that place value slot.

Question 9. Express in the standard form :
(a) 3,18,65,00,000
(b) \( 786.3 \times 10^4 \)
(c) 5,00,00,000
(b) 42634.7
(d) 4786.3460
Answer:
Standard form means writing a number as a value between 1 and 10 multiplied by a power of 10:
(a) \( 3,18,65,00,000 = 3.1865 \times 10^9 \)

(b) \( 786.3 \times 10^4 = 7.863 \times 10^2 \times 10^4 = 7.863 \times 10^6 \)

(c) \( 5,00,00,000 = 5 \times 10^7 \)

(b) \( 42634.7 = 4.26347 \times 10^4 \)

(d) \( 4786.3460 = 4.7863460 \times 10^3 \)
In simple words: Move the decimal point so there is only one non-zero number before it. Then count how many places you moved the point to get the power of 10.

Exam Tip: Count the shifts carefully from the original decimal point to the new point to get the exact power of 10.

Question 10. Write the numbers in the usual form :
(a) \( 4.83 \times 10^7 \)
(b) \( 3.64 \times 10^5 \)
(c) \( 7.3 \times 10^3 \)
Answer:
Move the decimal point to the right as many times as shown by the power of 10:
(a) \( 4.83 \times 10^7 = 48300000 \) (move the point 7 places to the right)

(b) \( 3.64 \times 10^5 = 364000 \) (move the point 5 places to the right)

(c) \( 7.3 \times 10^3 = 7300 \) (move the point 3 places to the right)
In simple words: To write a number in normal form, move the decimal point to the right. Add zeros at the end if you run out of numbers.

Exam Tip: Use pencil arches to trace the decimal jumps when moving the point to avoid making simple placement errors.