GSEB Class 7 Maths Solutions Chapter 13 Exponents and Powers InText Questions

Try These (Page 250)

Question 1. Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case.
Answer: The table below provides more examples where numbers are written in their exponential form, showing both the base and its exponent.

In simple words: Look at the number, then see how many times a smaller number multiplies itself to make that big number. The small number is the base, and how many times it multiplies is the exponent.

Exam Tip: To get full marks, ensure your factorization is correct and you clearly state both the base and the exponent for each exponential form.

Note:

1. \( x \times x \times x \times x = x^4 \) is read as 'x raised to the power 4' or '4th power of x'.

2. \( x^2y^5 \) is read as 'x squared into y raised to power 5'.

3. \( p^6q^3 \) is read as 'p raised to the power 6 into q cubed'.

Exam Tip: Remember the clear language used to express exponents in words; this precision is important in mathematics.

Try These (Page 251)

Question 1. Express:
(i) 729 as a power of 3
(ii) 128 as a power of 2
(iii) 343 as a power of 7
Answer:
(i) To express 729 as a power of 3, we repeatedly divide 729 by 3. We find that \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 3^6 \). Therefore, 729 is \( 3^6 \).
(ii) To express 128 as a power of 2, we repeatedly divide 128 by 2. We find that \( 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7 \). Therefore, 128 is \( 2^7 \).
(iii) To express 343 as a power of 7, we repeatedly divide 343 by 7. We find that \( 343 = 7 \times 7 \times 7 = 7^3 \). Therefore, 343 is \( 7^3 \).
In simple words: To write a number as a power of another, divide it by the base number again and again until you get 1. Count how many times you divided; that's your exponent.

Exam Tip: Show the step-by-step prime factorization for each number to clearly demonstrate how you arrived at the exponential form, as this is crucial for getting marks.

Try These (Page 254)

Question 1. Simplify and write in exponential form:
(i) \( 2^5 \times 2^3 \)
(ii) \( p^3 \times p^2 \)
(iii) \( 4^3 \times 4^2 \)
(iv) \( a^3 \times a^2 \times a^7 \)
(v) \( 5^3 \times 5^7 \times 5^{12} \)
(vi) \( (-4)^{100} \times (-4)^{20} \)
Answer: Since the rule is \( a^m \times a^n = a^{m+n} \), we can add the exponents when the bases are the same.
(i) For \( 2^5 \times 2^3 \): We add the powers: \( 2^{5+3} = 2^8 \).
(ii) For \( p^3 \times p^2 \): We add the powers: \( p^{3+2} = p^5 \).
(iii) For \( 4^3 \times 4^2 \): We add the powers: \( 4^{3+2} = 4^5 \).
(iv) For \( a^3 \times a^2 \times a^7 \): We add the powers: \( a^{3+2+7} = a^{12} \).
(v) For \( 5^3 \times 5^7 \times 5^{12} \): We add the powers: \( 5^{3+7+12} = 5^{22} \).
(vi) For \( (-4)^{100} \times (-4)^{20} \): We add the powers: \( (-4)^{100+20} = (-4)^{120} \).
In simple words: When you multiply numbers that have the same base but different powers, you just add the powers together. Keep the base as it is.

Exam Tip: Always remember that the law of exponents \( a^m \times a^n = a^{m+n} \) only applies when the bases are identical. Double-check your bases before adding exponents.

Note: The above rule is possible only for same bases. It is not true for different bases. Thus, \( 2^3 \times 3^2 \) will not obey this rule.

Exam Tip: Clearly understanding the conditions for applying exponent rules, such as having the same base for multiplication/division, prevents common errors.

Try These (Page 255)

Question 1. Simplify and write in exponential form: (example \( 11^6 \div 11^2 = 11^4 \) )
(i) \( 2^9 \div 2^3 \)
(ii) \( 10^8 \div 10^4 \)
(iii) \( 9^{11} \div 9^7 \)
(iv) \( 20^{15} \div 20^{13} \)
(v) \( 7^{13} \div 7^{10} \)
Answer: Since the rule is \( a^m \div a^n = a^{m-n} \), we can subtract the exponents when the bases are the same.
(i) For \( 2^9 \div 2^3 \): We subtract the powers: \( 2^{9-3} = 2^6 \).
(ii) For \( 10^8 \div 10^4 \): We subtract the powers: \( 10^{8-4} = 10^4 \).
(iii) For \( 9^{11} \div 9^7 \): We subtract the powers: \( 9^{11-7} = 9^4 \).
(iv) For \( 20^{15} \div 20^{13} \): We subtract the powers: \( 20^{15-13} = 20^2 \).
(v) For \( 7^{13} \div 7^{10} \): We subtract the powers: \( 7^{13-10} = 7^3 \).
In simple words: When you divide numbers that have the same base, you simply subtract the power of the bottom number from the power of the top number. Keep the base the same.

Exam Tip: Pay close attention to the order of subtraction; it's always the exponent of the divisor subtracted from the exponent of the dividend. A common mistake is subtracting in the wrong order.

Try These (Page 255)

Question 1. Simplify and write the answer in exponential form:
(i) \( (6^2)^4 \)
(ii) \( (2^2)^{100} \)
(iii) \( (7^{50})^2 \)
(iv) \( (5^3)^7 \)
Answer: Since the rule is \( (a^m)^n = a^{m \times n} \), we multiply the exponents when a power is raised to another power.
(i) For \( (6^2)^4 \): We multiply the powers: \( 6^{2 \times 4} = 6^8 \).
(ii) For \( (2^2)^{100} \): We multiply the powers: \( 2^{2 \times 100} = 2^{200} \).
(iii) For \( (7^{50})^2 \): We multiply the powers: \( 7^{50 \times 2} = 7^{100} \).
(iv) For \( (5^3)^7 \): We multiply the powers: \( 5^{3 \times 7} = 5^{21} \).
In simple words: When you have a number with a power, and that whole thing has another power outside the bracket, you just multiply the two powers together. The base stays the same.

Exam Tip: Make sure to distinguish this rule from \( a^m \times a^n \). This rule only applies when there is a power raised to *another* power, not when two separate exponential terms are multiplied.

Try These (Page 256)

Question 1. Put into another form using \( a^m \times b^m = (ab)^m \).
(i) \( 4^3 \times 2^3 \)
(ii) \( 2^5 \times b^5 \)
(iii) \( a^2 \times t^2 \)
(iv) \( 5^6 \times (-2)^6 \)
(v) \( (-2)^4 \times (-3)^4 \)
Answer: According to the rule \( a^m \times b^m = (ab)^m \), when terms have different bases but the same exponent, we can multiply the bases and keep the common exponent.
(i) For \( 4^3 \times 2^3 \): We combine the bases: \( (4 \times 2)^3 = 8^3 \).
(ii) For \( 2^5 \times b^5 \): We combine the bases: \( (2 \times b)^5 = (2b)^5 \).
(iii) For \( a^2 \times t^2 \): We combine the bases: \( (a \times t)^2 = (at)^2 \).
(iv) For \( 5^6 \times (-2)^6 \): We combine the bases: \( [5 \times (-2)]^6 = (-10)^6 \).
(v) For \( (-2)^4 \times (-3)^4 \): We combine the bases: \( [(-2) \times (-3)]^4 = [6]^4 \).
In simple words: If two different numbers each have the exact same power, you can first multiply those numbers together, and then put that same power on their product.

Exam Tip: This rule is useful for simplifying expressions. Remember it only works if the exponents are identical for all terms being multiplied.

Try These (Page 257)

Question 1. Put into another form using \( a^m \div b^m = \left(\frac{a}{b}\right)^m \).
(i) \( 4^5 \div 3^5 \)
(ii) \( 2^5 \div b^5 \)
(iii) \( (-2)^3 \div b^3 \)
(iv) \( p^4 \div q^4 \)
(v) \( 5^6 \div (-2)^6 \)
Answer: Based on the rule \( a^m \div b^m = \left(\frac{a}{b}\right)^m \), when terms have different bases but the same exponent and are being divided, we can divide the bases and keep the common exponent.
(i) For \( 4^5 \div 3^5 \): We combine the bases: \( \left(\frac{4}{3}\right)^5 \).
(ii) For \( 2^5 \div b^5 \): We combine the bases: \( \left(\frac{2}{b}\right)^5 \).
(iii) For \( (-2)^3 \div b^3 \): We combine the bases: \( \left(\frac{-2}{b}\right)^3 \).
(iv) For \( p^4 \div q^4 \): We combine the bases: \( \left(\frac{p}{q}\right)^4 \).
(v) For \( 5^6 \div (-2)^6 \): We combine the bases: \( \left(\frac{5}{-2}\right)^6 = \left(-\frac{5}{2}\right)^6 \).
In simple words: If you're dividing two numbers that have the same power, you can divide the numbers first, and then put that same power on the result.

Exam Tip: Make sure to enclose the division of bases in parentheses before applying the common exponent. This maintains the correct mathematical order of operations.

Try These (Page 261)

Question 1. Expand by expressing powers of 10 in the exponential form:
(i) 172
(ii) 5,643
(iii) 56,439
(iv) 1,76,428
Answer: To expand numbers using powers of 10, we break down each digit by its place value and multiply it by the corresponding power of 10. Remember that \( 10^0 = 1 \).
(i) For 172: \( 172 = (1 \times 100) + (7 \times 10) + (2 \times 1) \). This expands to \( 1 \times 10^2 + 7 \times 10^1 + 2 \times 10^0 \).
(ii) For 5,643: \( 5,643 = 5 \times 1000 + 6 \times 100 + 4 \times 10 + 3 \times 1 \). This expands to \( 5 \times 10^3 + 6 \times 10^2 + 4 \times 10^1 + 3 \times 10^0 \).
(iii) For 56,439: \( 56,439 = 5 \times 10000 + 6 \times 1000 + 4 \times 100 + 3 \times 10 + 9 \times 1 \). This expands to \( 5 \times 10^4 + 6 \times 10^3 + 4 \times 10^2 + 3 \times 10^1 + 9 \times 10^0 \).
(iv) For 1,76,428: \( 1,76,428 = 1 \times 100000 + 7 \times 10000 + 6 \times 1000 + 4 \times 100 + 2 \times 10 + 8 \times 1 \). This expands to \( 1 \times 10^5 + 7 \times 10^4 + 6 \times 10^3 + 4 \times 10^2 + 2 \times 10^1 + 8 \times 10^0 \).
In simple words: To expand a number using powers of 10, just take each digit, multiply it by its place value (like 10, 100, 1000), and write those place values as powers of 10. Then add them all up.

Exam Tip: Remember that \( 10^0 = 1 \) for the units place. Be careful with large numbers and make sure each digit corresponds to the correct power of 10 for its position.