CBSE Class 7 Mathematics Exponents And Powers Worksheet Set 08

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:
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. \((-1)^6 = +1\)
In simple words: Multiply the base number by itself as many times as the exponent indicates.
Exam Tip: A negative number raised to an odd power is negative, while a negative number raised to an even power is positive.

Question 2. Simplify:
a. \(3 \times 10^2\)
b. \(2^5 \times 5^3\)
c. \(0 \times 10^4\)
d. \(\left(\frac{3}{4}\right)^3\)
e. \(\left(-\frac{2}{3}\right)^4\)
Answer:
a. \(3 \times 10^2 = 3 \times 100 = 300\)
b. \(2^5 \times 5^3 = (2 \times 2 \times 2 \times 2 \times 2) \times (5 \times 5 \times 5) = 32 \times 125 = 4000\)
c. \(0 \times 10^4 = 0 \times 10000 = 0\)
d. \(\left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64}\)
e. \(\left(-\frac{2}{3}\right)^4 = \frac{(-2)^4}{3^4} = \frac{16}{81}\)
In simple words: Evaluate the powers first and then perform the multiplication or division.

Exam Tip: Any number multiplied by zero gives zero.

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. \(-5 \times -5 \times -5 \times -5\)
c. \(x \times x \times x \times x \times x \times a \times a \times b \times b \times b\)
d. \((-2) \times (-2) \times (-2) \times (-2) \times a \times a \times x \times a\)
Answer:
a. \(\left(-\frac{5}{7}\right)^4\)
b. \((-5)^4\)
c. \(x^5 a^2 b^3\)
d. \((-2)^4 a^3 x\)
In simple words: Count how many times each term is multiplied and write it as a power.

Exam Tip: Group identical bases together before writing in exponential form.

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:
a. \(36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2\)
b. \(675 = 3 \times 3 \times 3 \times 5 \times 5 = 3^3 \times 5^2\)
c. \(392 = 2 \times 2 \times 2 \times 7 \times 7 = 2^3 \times 7^2\)
d. \(864 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^5 \times 3^3\)
e. \(450 = 2 \times 3 \times 3 \times 5 \times 5 = 2 \times 3^2 \times 5^2\)
f. \(1800 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 = 2^3 \times 3^2 \times 5^2\)
In simple words: Find the prime factorization of the number using division and group identical prime factors with exponents.

Exam Tip: Always start dividing by the smallest prime number (2, 3, 5, etc.) systematically.

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:
(i) \(3^{6+5} = 3^{11}\)
(ii) \(7^{2 \times 3} \div 7^3 = 7^6 \div 7^3 = 7^{6-3} = 7^3\)
(iii) \(2^{20-5} = 2^{15}\)
(iv) \((2 \times 5)^4 = 10^4\)
(v) \((1 + 1)(1 + 1) = 2 \times 2 = 4\)
(vi) \(\left(\frac{7}{5}\right)^3\)
In simple words: Apply exponent laws: \(a^m \times a^n = a^{m+n}\), \(a^m \div a^n = a^{m-n}\), and \(a^0 = 1\).

Exam Tip: Memorize all standard laws of exponents before solving.

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:
(i) \(\frac{2^{15}}{2^{10}} = 2^{15-10} = 2^5\)
(ii) \((3^7)^3 = 3^{21}\)
(iii) \(\frac{2^{12} \times 2^8}{2^{12}} = \frac{2^{20}}{2^{12}} = 2^8\)
(iv) \(\frac{5^4}{5^4} \times x^{10-7} \times y^{5-4} = 1 \times x^3 \times y = x^3 y\)
(v) \(\left(\frac{2}{3} \times \frac{3}{5}\right)^5 = \left(\frac{2}{5}\right)^5\)
(vi) \(\frac{(3^2)^8 \times x^{10}}{(3^3)^4 \times x^6} = \frac{3^{16} \times x^{10}}{3^{12} \times x^6} = 3^{16-12} \times x^{10-6} = 3^4 x^4 = (3x)^4\)
(vii) Note that \(21 = 3 \times 7\) and \(91 = 7 \times 13\):
\(\frac{3^2 \times 7^8 \times 13^6}{(3 \times 7)^2 \times (7 \times 13)^3} = \frac{3^2 \times 7^8 \times 13^6}{3^2 \times 7^2 \times 7^3 \times 13^3} = \frac{3^2 \times 7^8 \times 13^6}{3^2 \times 7^5 \times 13^3} = 3^{2-2} \times 7^{8-5} \times 13^{6-3} = 7^3 \times 13^3\) (or \((7 \times 13)^3 = 91^3\))
(viii) Express \(10 = 2 \times 5\) and \(25 = 5^2\), and factor out \(5^n\):
\(\frac{2 \cdot 5 \cdot 5^{n+1} + 5^2 \cdot 5^n}{3 \cdot 5^{n+2} + 2 \cdot 5 \cdot 5^{n+1}} = \frac{2 \cdot 5^{n+2} + 5^{n+2}}{3 \cdot 5^{n+2} + 2 \cdot 5^{n+2}} = \frac{5^{n+2}(2 + 1)}{5^{n+2}(3 + 2)} = \frac{3}{5}\)
In simple words: Simplify numerator and denominator separately using exponent rules, then cancel common factors.

Exam Tip: Convert composite bases like 9, 27, 21, and 91 into prime bases (like 2, 3, 5, 7, 13) to easily apply exponent laws.

Question 7. Write the numbers in expanded forms:
a) 20068
b) 423719
c) 680071
d) 5004132
Answer:
a) \(2 \times 10^4 + 0 \times 10^3 + 0 \times 10^2 + 6 \times 10^1 + 8 \times 10^0\)
b) \(4 \times 10^5 + 2 \times 10^4 + 3 \times 10^3 + 7 \times 10^2 + 1 \times 10^1 + 9 \times 10^0\)
c) \(6 \times 10^5 + 8 \times 10^4 + 0 \times 10^3 + 0 \times 10^2 + 7 \times 10^1 + 1 \times 10^0\)
d) \(5 \times 10^6 + 0 \times 10^5 + 0 \times 10^4 + 4 \times 10^3 + 1 \times 10^2 + 3 \times 10^1 + 2 \times 10^0\)
In simple words: Multiply each digit by its respective place value expressed as a power of 10.

Exam Tip: Count digits from right to left starting with power 0 for the units place.

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:
(a) \(500000 + 40000 + 2000 + 3 = 542003\)
(b) \(9000000 + 80000 + 700 + 6 = 9080706\)
(c) \(30000 + 4000 + 5 = 34005\)
In simple words: Combine the place values to write the standard numeral, remembering to put 0 for missing powers.

Exam Tip: Be careful with missing powers (like \(10^5\) or \(10^1\) in part b); they represent zero in that place value position.

Question 9. Express in the standard form:
(a) 3,18,65,00,000
(b) 786.3
(c) 5,00,00,000
(d) 42634.7
(e) 4786.3460
Answer:
(a) \(3.1865 \times 10^9\)
(b) \(7.863 \times 10^2\)
(c) \(5.0 \times 10^7\)
(d) \(4.26347 \times 10^4\)
(e) \(4.786346 \times 10^3\)

In simple words: Express the number as a decimal between 1 and 10 multiplied by a power of 10.

Exam Tip: Moving the decimal point to the left makes the exponent of 10 positive.

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:
(a) \(48,300,000\)
(b) \(364,000\)
(c) \(7,300\)
In simple words: Shift the decimal point to the right according to the positive power of 10.

Exam Tip: Add zeros as placeholders if there are not enough digits after the decimal point.