GSEB Class 8 Maths Solutions Chapter 12 Exponents and Powers InText Questions

Try These (Page 194)

Question 1. Find the multiplicative inverse of the following?
1. \( 2^{-4} \)
2. \( 10^{-5} \)
3. \( 7^{-2} \)
4. \( 5^{-3} \)
5. \( 100^{-100} \)
Answer:
1. The reciprocal of \( 2^{-4} \) is \( 2^4 \).
2. For \( 10^{-5} \), its inverse in multiplication is \( 10^5 \).
3. The inverse for \( 7^{-2} \) is \( 7^2 \).
4. The reciprocal for \( 5^{-3} \) becomes \( 5^3 \).
5. For \( 10^{-100} \), the inverse is \( 10^{100} \).
In simple words: To find the multiplicative inverse of a number with a negative exponent, simply change the sign of the exponent to positive. This gives you the reciprocal of the original number.

Exam Tip: Remember that the multiplicative inverse (or reciprocal) of \( a^{-n} \) is always \( a^n \). This rule helps to quickly solve problems involving negative exponents.

Try These (Page 194)

Question 1. Expand the following number using exponents?
(i) 1025.63
(ii) 1256.249
Answer:

Exam Tip: Remember that \( 10^0 = 1 \). For decimal places, the first digit after the decimal is multiplied by \( 10^{-1} \), the second by \( 10^{-2} \), and so on.

Try These (Page 195)

Question 1. Simplify and write in exponential form?
1. \( (-2)^3 \times (-2)^{-4} \)
2. \( p^3 \times p^{-10} \)
3. \( 3^2 \times 3^{-5} \times 3^6 \)
Answer:
1. \( (-2)^3 \times (-2)^{-4} = (-2)^{3+(-4)} = (-2)^{-1} \) or \( \frac{1}{(-2)^1} \)
2. \( p^3 \times p^{-10} = p^{3+(-10)} = p^{-7} \) or \( \frac{1}{p^7} \)
3. \( 3^2 \times 3^{-5} \times 3^6 = 3^{2+(-5)+6} = 3^{8-5} = 3^3 \)
In simple words: When multiplying numbers that have the same base but different exponents, you simply add the exponents together. If the result is a negative exponent, you can write it as 1 divided by the base with a positive exponent.

Exam Tip: Always remember the basic rule for multiplying powers with the same base: \( a^m \times a^n = a^{m+n} \). Pay close attention to negative exponents in the sum.

Law II: \( \frac{a^m}{a^n} = a^{m-n} \)
Example: \( 5^{-1} \div 5^{-2} = 5^{-1-(-2)} = 5^{-1+2} = 5^1 \) or \( 5 \)

Law III: \( (a^m)^n = a^{mn} \)
Example: \( (9^{-1})^{-3} = 9^{(-1) \times (-3)} = 9^3 \)

Law IV: \( a^m \times b^m = (ab)^m \)
Example: \( 2^{-4} \times 3^{-4} = (2 \times 3)^{-4} = 6^{-4} \) or \( \frac{1}{6^4} \)

Law V: \( \frac{a^m}{b^m} = (\frac{a}{b})^m \)
Example: \( \frac{3^{-5}}{7^{-5}} = (\frac{3}{7})^{-5} \) or \( (\frac{7}{3})^5 \)

Law VI: \( a^0 = 1 \)
Example:
(i) \( (-38)^0 = 1 \)
(ii) \( (32456)^0 = 1 \)

Try These (Page 199)

Question 1. Write the following numbers in standard form?
1. 0.000000564
2. 0.0000021
3. 15240000
Answer:
1. \( 0.000000564 = \frac{564}{1,000,000,000} = \frac{5.64 \times 10^2}{10^9} = \frac{5.64}{10^7} = 5.64 \times 10^{-7} \)
2. \( 0.0000021 = \frac{21}{10,000,000} = \frac{2.1 \times 10}{10^7} = \frac{2.1}{10^6} = 2.1 \times 10^{-6} \)
3. \( 15240000 = 1.524 \times 10,000,000 = 1.524 \times 10^7 \)
In simple words: To write a number in standard form, move the decimal point so that there is only one non-zero digit before it. The number of places you moved the decimal gives you the power of 10. For small numbers, the exponent is negative, and for large numbers, it is positive.

Exam Tip: For standard form, the number \( k \) must always be between 1 and 10 (including 1 but not 10). The exponent \( n \) indicates how many places the decimal point was moved and in which direction (negative for left, positive for right).

Question 2. Write all the facts given in the standard form?
Answer: A number is expressed in standard form when it is written as \( k \times 10^n \), where \( 1 \le k < 10 \) and \( n \) is an integer. This means a number is shown as a product of a value between 1 and 10 and a power of 10.
For example, let's compare the size of a red blood cell, which is \( 0.000007 \text{ m} \), to that of a plant cell, which is \( 0.0000129 \text{ m} \).
Size of red blood cell \( = 0.000007 \text{ m} = \frac{7}{1,000,000} \text{ m} = 7 \times 10^{-6} \text{ m} \)
Size of the plant cell \( = 0.0000129 \text{ m} = \frac{129}{10,000,000} \text{ m} = 1.29 \times 10^{-5} \text{ m} \)
To compare their sizes, we find the ratio:
\( \frac{\text{Size of red blood cell}}{\text{Size of plant cell}} = \frac{7 \times 10^{-6} \text{ m}}{1.29 \times 10^{-5} \text{ m}} = \frac{7}{1.29} \times 10^{-6 - (-5)} \)
\( = \frac{7}{1.29} \times 10^{-1} = \frac{7 \times 10^{-1}}{1.29} = \frac{0.7}{1.29} \)
Approximating, \( \frac{0.7}{1.3} \approx \frac{1}{2} \)
Thus, the size of a red blood cell is about half the size of the plant cell.
In simple words: Standard form writes numbers as a value between 1 and 10 multiplied by a power of 10. For comparing cell sizes, we convert both to standard form and then divide them to see their ratio. The red blood cell is approximately half the size of the plant cell.

Exam Tip: When converting very small numbers to standard form, the exponent will be negative. For comparing quantities, express both in standard form first, then use division to find their ratio, which makes the comparison clearer.