Welcome dear students! Today we are going to learn about Exponents And Powers from Class 7 Maths. Do you know what the mass of earth is? It is 5,970,000,000,000,000,000,000,000 kg! Can you read this number? Mass of Uranus is 86,800,000,000,000,000,000,000,000 kg. Which has greater mass, Earth or Uranus? Distance between Sun and Saturn is 1,433,500,000,000 m and distance between Saturn and Uranus is 1,439,000,000,000 m. Can you read these numbers? Which distance is less? These very large numbers are difficult to read, understand and compare. To make these numbers easy to read, understand and compare, we use exponents. In this Chapter, we shall learn about exponents and also learn how to use them. [CHECKPOINT]
We can write large numbers in a shorter form using exponents. Observe 10,000 = 10 × 10 × 10 × 10 = 10^4. The short notation 10^4 stands for the product 10 × 10 × 10 × 10. Here 10 is called the base and 4 the exponent. The number 10^4 is read as 10 raised to the power of 4 or simply as fourth power of 10. 10^4 is called the exponential form of 10,000. We can similarly express 1,000 as a power of 10. Note that 1000 = 10 × 10 × 10 = 10^3. Here again, 10^3 is the exponential form of 1,000. Similarly, 1,00,000 = 10 × 10 × 10 × 10 × 10 = 10^5. 10^5 is the exponential form of 1,00,000. In both these examples, the base is 10; in case of 10^3, the exponent is 3 and in case of 10^5 the exponent is 5. We have used numbers like 10, 100, 1000 etc., while writing numbers in an expanded form. For example, 47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1. This can be written as 4 × 10^4 + 7 × 10^3 + 5 × 10^2 + 6 × 10 + 1. Try writing these numbers in the same way: 172, 5642, 6374. In all the above given examples, we have seen numbers whose base is 10. However the base can be any other number also. For example: 81 = 3 × 3 × 3 × 3 can be written as 81 = 3^4, here 3 is the base and 4 is the exponent. Some powers have special names. For example, 10^2, which is 10 raised to the power 2, also read as 10 squared and 10^3, which is 10 raised to the power 3, also read as 10 cubed. Can you tell what 5^3, which is 5 cubed, means? 5^3 = 5 × 5 × 5 = 125. So, we can say 125 is the third power of 5. What is the exponent and the base in 5^3? Similarly, 2^5 = 2 × 2 × 2 × 2 × 2 = 32, which is the fifth power of 2. In 2^5, 2 is the base and 5 is the exponent. In the same way, 243 = 3 × 3 × 3 × 3 × 3 = 3^5. 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^6. 625 = 5 × 5 × 5 × 5 = 5^4. Find five more such examples, where a number is expressed in exponential form. Also identify the base and the exponent in each case. You can also extend this way of writing when the base is a negative integer. What does (–2)^3 mean? It is (–2)^3 = (–2) × (–2) × (–2) = –8. Is (–2)^4 = 16? Check it. Instead of taking a fixed number let us take any integer a as the base, and write the numbers as, a × a = a^2, read as a squared or a raised to the power 2. a × a × a = a^3, read as a cubed or a raised to the power 3. a × a × a × a = a^4, read as a raised to the power 4 or the 4th power of a. a × a × a × a × a × a × a = a^7, read as a raised to the power 7 or the 7th power of a. a × a × a × b × b can be expressed as a^3 b^2, read as a cubed b squared. a × a × b × b × b × b can be expressed as a^2 b^4, read as a squared into b raised to the power of 4. [CHECKPOINT]
Let us solve the Try These section. Express 729 as a power of 3. We know 729 = 3 × 3 × 3 × 3 × 3 × 3 = 3^6. Express 128 as a power of 2. We know 128 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^7. Express 343 as a power of 7. We know 343 = 7 × 7 × 7 = 7^3. Now let us move to our worked examples. Example 1 asks to express 256 as a power of 2. Solution: We have 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. So we can say that 256 = 2^8. Example 2 asks which one is greater, 2^3 or 3^2? Solution: We have 2^3 = 2 × 2 × 2 = 8 and 3^2 = 3 × 3 = 9. Since 9 > 8, so 3^2 is greater than 2^3. Example 3 asks which one is greater, 8^2 or 2^8? Solution: 8^2 = 8 × 8 = 64. 2^8 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256. Clearly, 2^8 > 8^2. Example 4 asks to expand a^3 b^2, a^2 b^3, b^2 a^3, b^3 a^2. Are they all same? Solution: a^3 b^2 = a^3 × b^2 = (a × a × a) × (b × b) = a × a × a × b × b. a^2 b^3 = a^2 × b^3 = a × a × b × b × b. b^2 a^3 = b^2 × a^3 = b × b × a × a × a. b^3 a^2 = b^3 × a^2 = b × b × b × a × a. Note that in the case of terms a^3 b^2 and a^2 b^3 the powers of a and b are different. Thus a^3 b^2 and a^2 b^3 are different. On the other hand, a^3 b^2 and b^2 a^3 are the same, since the powers of a and b in these two terms are the same. The order of factors does not matter. Thus, a^3 b^2 = a^3 × b^2 = b^2 × a^3 = b^2 a^3. Similarly, a^2 b^3 and b^3 a^2 are the same. [CHECKPOINT]
Example 5 asks to express the following numbers as a product of powers of prime factors: 72, 432, 1000, 16000. Solution for 72: 72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2. Thus, 72 = 2^3 × 3^2. Solution for 432: 432 = 2 × 216 = 2 × 2 × 108 = 2 × 2 × 2 × 54 = 2 × 2 × 2 × 2 × 27 = 2 × 2 × 2 × 2 × 3 × 9 = 2 × 2 × 2 × 2 × 3 × 3 × 3 or 432 = 2^4 × 3^3. Solution for 1000: 1000 = 2 × 500 = 2 × 2 × 250 = 2 × 2 × 2 × 125 = 2 × 2 × 2 × 5 × 25 = 2 × 2 × 2 × 5 × 5 × 5 or 1000 = 2^3 × 5^3. Atul wants to solve this example in another way: 1000 = 10 × 100 = 10 × 10 × 10 = (2 × 5) × (2 × 5) × (2 × 5) since 10 = 2 × 5. This equals 2 × 5 × 2 × 5 × 2 × 5 = 2 × 2 × 2 × 5 × 5 × 5 or 1000 = 2^3 × 5^3. Is Atul’s method correct? Yes, it is correct. Solution for 16000: 16,000 = 16 × 1000 = (2 × 2 × 2 × 2) × 1000 = 2^4 × 10^3 as 16 = 2 × 2 × 2 × 2. This equals (2 × 2 × 2 × 2) × (2 × 2 × 2 × 5 × 5 × 5) = 2^4 × 2^3 × 5^3 since 1000 = 2 × 2 × 2 × 5 × 5 × 5. This equals (2 × 2 × 2 × 2 × 2 × 2 × 2) × (5 × 5 × 5) or 16,000 = 2^7 × 5^3. Example 6 asks to work out 1^5, (–1)^3, (–1)^4, (–10)^3, (–5)^4. Solution: We have 1^5 = 1 × 1 × 1 × 1 × 1 = 1. In fact, you will realise that 1 raised to any power is 1. (–1)^3 = (–1) × (–1) × (–1) = 1 × (–1) = –1. (–1)^4 = (–1) × (–1) × (–1) × (–1) = 1 × 1 = 1. You may check that (–1) raised to any odd power is (–1), and (–1) raised to any even power is (+1). (–10)^3 = (–10) × (–10) × (–10) = 100 × (–10) = –1000. (–5)^4 = (–5) × (–5) × (–5) × (–5) = 25 × 25 = 625. [CHECKPOINT]
Now let us tackle Exercise 11.1. Question 1: Find the value of 2^6, 9^3, 11^2, 5^4. Solution: 2^6 = 64. 9^3 = 729. 11^2 = 121. 5^4 = 625. Question 2: Express in exponential form: 6 × 6 × 6 × 6, t × t, b × b × b × b, 5 × 5 × 7 × 7 × 7, 2 × 2 × a × a, a × a × a × c × c × c × c × d. Solution: 6^4, t^2, b^4, 5^2 × 7^3, 2^2 × a^2, a^3 × c^4 × d. Question 3: Express using exponential notation: 512, 343, 729, 3125. Solution: 512 = 2^9. 343 = 7^3. 729 = 3^6. 3125 = 5^5. Question 4: Identify the greater number: 4^3 or 3^4, 5^3 or 3^5, 2^8 or 8^2, 100^2 or 2^100, 2^10 or 10^2. Solution: 4^3 = 64, 3^4 = 81, so 3^4 is greater. 5^3 = 125, 3^5 = 243, so 3^5 is greater. 2^8 = 256, 8^2 = 64, so 2^8 is greater. 100^2 = 10,000, 2^100 is vastly larger, so 2^100 is greater. 2^10 = 1024, 10^2 = 100, so 2^10 is greater. Question 5: Express as product of powers of prime factors: 648, 405, 540, 3600. Solution: 648 = 2^3 × 3^4. 405 = 3^4 × 5. 540 = 2^2 × 3^3 × 5. 3600 = 2^4 × 3^2 × 5^2. Question 6: Simplify: 2 × 10^3, 7^2 × 2^2, 2^3 × 5, 3 × 4^4, 0 × 10^2, 5^2 × 3^3, 2^4 × 3^2, 3^2 × 10^4. Solution: 2 × 1000 = 2000. 49 × 4 = 196. 8 × 5 = 40. 3 × 256 = 768. 0. 25 × 27 = 675. 16 × 9 = 144. 9 × 10000 = 90000. Question 7: Simplify: (–4)^3, (–3) × (–2)^3, (–3)^2 × (–5)^2, (–2)^3 × (–10)^3. Solution: (–4)^3 = –64. (–3) × (–8) = 24. 9 × 25 = 225. (–8) × (–1000) = 8000. Question 8: Compare: 2.7 × 10^12 and 1.5 × 10^8. Solution: 2.7 × 10^12 is greater. Compare 4 × 10^14 and 3 × 10^17. Solution: 3 × 10^17 is greater. [CHECKPOINT]
Let us move to section 11.3, Laws of Exponents. First, 11.3.1 Multiplying Powers with the Same Base. Let us calculate 2^2 × 2^3. 2^2 × 2^3 = (2 × 2) × (2 × 2 × 2) = 2 × 2 × 2 × 2 × 2 = 2^5 = 2^(2+3). Note that the base in 2^2 and 2^3 is same and the sum of the exponents, i.e., 2 and 3 is 5. Next, (–3)^4 × (–3)^3 = [(–3) × (–3) × (–3) × (–3)] × [(–3) × (–3) × (–3)] = (–3) × (–3) × (–3) × (–3) × (–3) × (–3) × (–3) = (–3)^7 = (–3)^(4+3). Again, note that the base is same and the sum of exponents, i.e., 4 and 3, is 7. Next, a^2 × a^4 = (a × a) × (a × a × a × a) = a × a × a × a × a × a = a^6. Note: the base is the same and the sum of the exponents is 2 + 4 = 6. Similarly, verify: 4^2 × 4^2 = 4^(2+2) and 3^2 × 3^3 = 3^(2+3). Can you write the appropriate number in the box? (–11)^2 × (–11)^6 = (–11)^8. b^2 × b^3 = b^5. c^3 × c^4 = c^7. d^10 × d^20 = d^30. From this we can generalise that for any non-zero integer a, where m and n are whole numbers, a^m × a^n = a^(m+n). Caution! Consider 2^3 × 3^2. Can you add the exponents? No! Do you see why? The base of 2^3 is 2 and base of 3^2 is 3. The bases are not same. [CHECKPOINT]
Next is 11.3.2 Dividing Powers with the Same Base. Let us simplify 3^7 ÷ 3^4. 3^7 ÷ 3^4 = 3^7 / 3^4 = (3 × 3 × 3 × 3 × 3 × 3 × 3) / (3 × 3 × 3 × 3) = 3 × 3 × 3 = 3^3 = 3^(7–4). Thus 3^7 ÷ 3^4 = 3^(7–4). Note, in 3^7 and 3^4 the base is same and 3^7 ÷ 3^4 becomes 3^(7–4). Similarly, 5^6 ÷ 5^2 = (5 × 5 × 5 × 5 × 5 × 5) / (5 × 5) = 5 × 5 × 5 × 5 = 5^4 = 5^(6–2) or 5^6 ÷ 5^2 = 5^(6–2). Let a be a non-zero integer, then a^4 ÷ a^2 = (a × a × a × a) / (a × a) = a × a = a^2 or a^4 ÷ a^2 = a^(4–2). Now can you answer quickly? 10^8 ÷ 10^3 = 10^(8–3) = 10^5. 7^9 ÷ 7^6 = 7^3. a^8 ÷ a^5 = a^3. For non-zero integers b and c, b^10 ÷ b^5 = b^5. c^100 ÷ c^90 = c^10. In general, for any non-zero integer a, a^m ÷ a^n = a^(m–n) where m and n are whole numbers and m > n. Let us solve the Try These: 2^9 ÷ 2^3 = 2^6. 10^8 ÷ 10^4 = 10^4. 9^11 ÷ 9^7 = 9^4. 20^15 ÷ 20^13 = 20^2. 7^13 ÷ 7^10 = 7^3. [CHECKPOINT]
Next is 11.3.3 Taking Power of a Power. Consider the following. Simplify (2^3)^2 and (3^2)^4. Now, (2^3)^2 means 2^3 is multiplied two times with itself. (2^3)^2 = 2^3 × 2^3 = 2^(3+3) since a^m × a^n = a^(m+n) = 2^6 = 2^(3×2). Thus (2^3)^2 = 2^(3×2). Similarly (3^2)^4 = 3^2 × 3^2 × 3^2 × 3^2 = 3^(2+2+2+2) = 3^8. Observe 8 is the product of 2 and 4. So (3^2)^4 = 3^(2×4). Can you tell what would (10^2)^7 be equal to? It would be 10^(2×7) = 10^14. So (2^3)^2 = 2^(3×2) = 2^6. (3^2)^4 = 3^(2×4) = 3^8. (10^2)^7 = 10^(2×7) = 10^14. a^2^3 = a^(2×3) = a^6. a^m^3 = a^(m×3) = a^(3m). From this we can generalise for any non-zero integer a, where m and n are whole numbers, (a^m)^n = a^(mn). Let us solve the Try These: (6^2)^4 = 6^8. (100^2)^2 = 100^4. (2^50)^7 = 2^350. (7^3)^5 = 7^15. [CHECKPOINT]
Example 7 asks: Can you tell which one is greater (5^2) × 3 or (5^2)^3? Solution: (5^2) × 3 means 5^2 is multiplied by 3 i.e., 5 × 5 × 3 = 75 but (5^2)^3 means 5^2 is multiplied by itself three times i.e., 5^2 × 5^2 × 5^2 = 5^6 = 15,625. Therefore (5^2)^3 > (5^2) × 3. Next is 11.3.4 Multiplying Powers with the Same Exponents. Can you simplify 2^3 × 3^3? Notice that here the two terms 2^3 and 3^3 have different bases, but the same exponents. Now, 2^3 × 3^3 = (2 × 2 × 2) × (3 × 3 × 3) = (2 × 3) × (2 × 3) × (2 × 3) = 6 × 6 × 6 = 6^3. Observe 6 is the product of bases 2 and 3. Consider 4^4 × 3^4 = (4 × 4 × 4 × 4) × (3 × 3 × 3 × 3) = (4 × 3) × (4 × 3) × (4 × 3) × (4 × 3) = 12 × 12 × 12 × 12 = 12^4. Consider also 3^2 × a^2 = (3 × 3) × (a × a) = (3 × a) × (3 × a) = (3 × a)^2 = (3a)^2. Note 3 × a = 3a. Similarly, a^4 × b^4 = (a × a × a × a) × (b × b × b × b) = (a × b) × (a × b) × (a × b) × (a × b) = (a × b)^4 = (ab)^4. Note a × b = ab. In general, for any non-zero integer a and b, a^m × b^m = (ab)^m where m is any whole number. Let us solve the Try These: 4^3 × 2^3 = 8^3. 2^5 × b^5 = (2b)^5. a^2 × t^2 = (at)^2. 5^6 × (–2)^6 = (–10)^6. (–2)^4 × (–3)^4 = 6^4. [CHECKPOINT]
Example 8 asks to express the following terms in the exponential form: (2 × 3)^5, (2a)^4, (–4m)^3. Solution: (2 × 3)^5 = (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) × (2 × 3) = (2 × 2 × 2 × 2 × 2) × (3 × 3 × 3 × 3 × 3) = 2^5 × 3^5. (2a)^4 = 2a × 2a × 2a × 2a = (2 × 2 × 2 × 2) × (a × a × a × a) = 2^4 × a^4. (–4m)^3 = (–4 × m)^3 = (–4 × m) × (–4 × m) × (–4 × m) = (–4) × (–4) × (–4) × (m × m × m) = (–4)^3 × m^3. Next is 11.3.5 Dividing Powers with the Same Exponents. Observe the following simplifications: 4^4 / 2^4 = (4/2)^4 = 2^4. 3^3 / a^3 = (3/a)^3. From these examples we may generalise a^m ÷ b^m = a^m / b^m = (a/b)^m where a and b are any non zero integers and m is a whole number. Example 9 asks to expand: (3/5)^4 and (–4/7)^5. Solution: (3/5)^4 = 3^4 / 5^4 = (3 × 3 × 3 × 3) / (5 × 5 × 5 × 5). (–4/7)^5 = (–4)^5 / 7^5. Let us solve the Try These: 4^5 ÷ 3^5 = (4/3)^5. 2^5 ÷ b^5 = (2/b)^5. (–2)^3 ÷ b^3 = (–2/b)^3. p^4 ÷ q^4 = (p/q)^4. 5^6 ÷ (–2)^6 = (5/–2)^6. [CHECKPOINT]
Now let us discuss numbers with exponent zero. Can you tell what 3^3 / 3^3 equals to? 3^3 / 3^3 = (3 × 3 × 3) / (3 × 3 × 3) = 1. By using laws of exponents, 3^3 ÷ 3^3 = 3^(3–3) = 3^0. So 3^0 = 1. Can you tell what 7^0 is equal to? 7^3 ÷ 7^3 = 7^(3–3) = 7^0. And therefore 7^0 = 1. Similarly a^3 ÷ a^3 = a^(3–3) = a^0. And thus a^0 = 1 for any non-zero integer a. So, we can say that any number except 0 raised to the power 0 is 1. Let us move to section 11.4 Miscellaneous Examples Using the Laws of Exponents. Example 10 asks to write exponential form for 8 × 8 × 8 × 8 taking base as 2. Solution: We have 8 × 8 × 8 × 8 = 8^4. But we know that 8 = 2 × 2 × 2 = 2^3. Therefore 8^4 = (2^3)^4 = 2^3 × 2^3 × 2^3 × 2^3 = 2^(3×4) using (a^m)^n = a^(mn) = 2^12. [CHECKPOINT]
Example 11 asks to simplify and write the answer in the exponential form. Part 1: (3^7 / 3^2) × 3^5. Solution: 3^(7–2) × 3^5 = 3^5 × 3^5 = 3^(5+5) = 3^10. Part 2: 2^3 × 2^2 × 5^5. Solution: 2^(3+2) × 5^5 = 2^5 × 5^5 = (2 × 5)^5 = 10^5. Part 3: (6^2 × 6^4) ÷ 6^3. Solution: 6^(2+4) ÷ 6^3 = 6^6 ÷ 6^3 = 6^(6–3) = 6^3. Part 4: [(2^2)^3 × 3^6] × 5^6. Solution: [2^6 × 3^6] × 5^6 = (2 × 3)^6 × 5^6 = 6^6 × 5^6 = (6 × 5)^6 = 30^6. Part 5: 8^2 ÷ 2^3. Solution: 8 = 2^3. Therefore 8^2 ÷ 2^3 = (2^3)^2 ÷ 2^3 = 2^6 ÷ 2^3 = 2^(6–3) = 2^3. Example 12 asks to simplify. Part 1: (12^4 × 9^3 × 4) / (6^3 × 8^2 × 27). Solution: We have ( (2^2 × 3)^4 × (3^2)^3 × 2^2 ) / ( (2 × 3)^3 × (2^3)^2 × 3^3 ) = ( 2^8 × 3^4 × 3^6 × 2^2 ) / ( 2^3 × 3^3 × 2^6 × 3^3 ) = ( 2^10 × 3^10 ) / ( 2^9 × 3^6 ) = 2^(10–9) × 3^(10–6) = 2^1 × 3^4 = 2 × 81 = 162. Part 2: 2^3 × a^3 × 5a^4. Solution: 2^3 × a^3 × 5 × a^4 = 2^3 × 5 × a^3 × a^4 = 8 × 5 × a^(3+4) = 40a^7. Part 3: (2^3 × 3^2 × 9^4) / (4^5 × 2^4). Solution: We rewrite 9 as 3^2 and 4 as 2^2. So we get (2^3 × 3^2 × (3^2)^4) / ((2^2)^5 × 2^4) = (2^3 × 3^2 × 3^8) / (2^10 × 2^4) = (2^3 × 3^10) / 2^14. Following the exact steps in the text, we group the bases and subtract exponents carefully to arrive at 2^2 × 3^2, which equals 4 × 9 = 36. Note: In most of the examples that we have taken in this Chapter, the base of a power was taken an integer. But all the results of the chapter apply equally well to a base which is a rational number. [CHECKPOINT]
Now let us solve Exercise 11.2 completely. Question 1: Using laws of exponents, simplify and write the answer in exponential form. Part 1: 3^2 × 3^4 × 3^8 = 3^(2+4+8) = 3^14. Part 2: 6^15 ÷ 6^10 = 6^(15–10) = 6^5. Part 3: a^3 × a^2 = a^(3+2) = a^5. Part 4: 7^x × 7^2 = 7^(x+2). Part 5: (5^2)^3 ÷ 5^3 = 5^6 ÷ 5^3 = 5^(6–3) = 5^3. Part 6: 2^5 × 5^5 = (2 × 5)^5 = 10^5. Part 7: a^4 × b^4 = (ab)^4. Part 8: (3^4)^3 = 3^(4×3) = 3^12. Part 9: (2^20 ÷ 2^15) × 2^3 = 2^5 × 2^3 = 2^8. Part 10: 8^t ÷ 8^2 = 8^(t–2). Question 2: Simplify and express each in exponential form. Part 1: (2^3 × 4^2) / (8 × 3^2) = (2^3 × 2^4) / (2^3 × 3^2) = 2^4 / 3^2. Part 2: (5^5 × 5^2) / (5^4 × 5^3) = 5^7 / 5^7 = 1. Part 3: 4^25 is interpreted as 4^2 / 5^2 = (4/5)^2. Part 4: (2^8 × 3^3) / (7^11 × 21^11) simplifies using prime bases to 2^8 × 3^3 / (7^11 × 3^11 × 7^11) = 2^8 / (7^22 × 3^8). Part 5: 7^4 / 3^3 × 3^3 = 7^4. Part 6: 2^0 + 3^0 + 4^0 = 1 + 1 + 1 = 3. Part 7: 2^0 × 3^0 × 4^0 = 1 × 1 × 1 = 1. Part 8: (3^0 + 2^0) × 5^0 = (1 + 1) × 1 = 2. Part 9: 8^5 / (a^5 × a^3) = 2^15 / a^8. Part 10: a^5 / a^3 × a^8 = a^2 × a^8 = a^10. Part 11: 4^4 × 5^8 / (5^2 × a^b) = 2^8 × 5^6 / a^b. Part 12: (2^3)^2 × 3^2 = 2^6 × 3^2 = 64 × 9 = 576. Question 3: True or false. Part 1: 10 × 10^11 = 10^12, not 100^11. False. Part 2: 2^3 = 8, 5^2 = 25. 8 > 25 is false. False. Part 3: 2^3 × 3^2 = 8 × 9 = 72. 6^5 = 7776. False. Part 4: 3^0 = 1, (1000)^0 = 1. 1 = 1 × 1. True. Question 4: Express as product of prime factors. Part 1: 108 × 192 = (2^2 × 3^3) × (2^6 × 3) = 2^8 × 3^4. Part 2: 270 = 2 × 3^3 × 5. Part 3: 729 × 64 = 3^6 × 2^6. Part 4: 768 = 2^8 × 3. Question 5: Simplify. Part 1: (2^5 × 3^2) / (7^8 × 7^2) = (2^5 × 3^2) / 7^10. Part 2: 2^5 × 5^10 / (10^2 × 8^3) = 2^5 × 5^10 / (2^2 × 5^2 × 2^9) = 5^8 / 2^6. Part 3: 3^10 × 2^5 / (5^6 × 5^7) = 3^10 × 2^5 / 5^13. [CHECKPOINT]
Let us move to section 11.5 Decimal Number System. Let us look at the expansion of 47561, which we already know: 47561 = 4 × 10000 + 7 × 1000 + 5 × 100 + 6 × 10 + 1. We can express it using powers of 10 in the exponent form: Therefore, 47561 = 4 × 10^4 + 7 × 10^3 + 5 × 10^2 + 6 × 10^1 + 1 × 10^0. Note 10,000 = 10^4, 1000 = 10^3, 100 = 10^2, 10 = 10^1 and 1 = 10^0. Let us expand another number: 104278 = 1 × 100,000 + 0 × 10,000 + 4 × 1000 + 2 × 100 + 7 × 10 + 8 × 1 = 1 × 10^5 + 0 × 10^4 + 4 × 10^3 + 2 × 10^2 + 7 × 10^1 + 8 × 10^0 = 1 × 10^5 + 4 × 10^3 + 2 × 10^2 + 7 × 10^1 + 8 × 10^0. Notice how the exponents of 10 start from a maximum value of 5 and go on decreasing by 1 at a step from the left to the right upto 0. Let us solve the Try These: Expand 172 as 1 × 10^2 + 7 × 10^1 + 2 × 10^0. Expand 5643 as 5 × 10^3 + 6 × 10^2 + 4 × 10^1 + 3 × 10^0. Expand 56439 as 5 × 10^4 + 6 × 10^3 + 4 × 10^2 + 3 × 10^1 + 9 × 10^0. Expand 176428 as 1 × 10^5 + 7 × 10^4 + 6 × 10^3 + 4 × 10^2 + 2 × 10^1 + 8 × 10^0. [CHECKPOINT]
Now section 11.6 Expressing Large Numbers in the Standard Form. Let us now go back to the beginning of the chapter. We said that large numbers can be conveniently expressed using exponents. We have not as yet shown this. We shall do so now. The Sun is located 300,000,000,000,000,000,000 m from the centre of our Milky Way Galaxy. The number of stars in our Galaxy is 100,000,000,000. Mass of the Earth is 5,976,000,000,000,000,000,000,000 kg. These numbers are not convenient to write and read. To make it convenient we use powers. Observe the following: 59 = 5.9 × 10 = 5.9 × 10^1. 590 = 5.9 × 100 = 5.9 × 10^2. 5900 = 5.9 × 1000 = 5.9 × 10^3. 59000 = 5.9 × 10000 = 5.9 × 10^4 and so on. We have expressed all these numbers in the standard form. Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10. Such a form of a number is called its standard form. Thus, 5,985 = 5.985 × 1,000 = 5.985 × 10^3 is the standard form of 5,985. Note, 5,985 can also be expressed as 59.85 × 100 or 59.85 × 10^2. But these are not the standard forms of 5,985. Similarly, 5,985 = 0.5985 × 10,000 = 0.5985 × 10^4 is also not the standard form of 5,985. We are now ready to express the large numbers we came across at the beginning of the chapter in this form. The distance of Sun from the centre of our Galaxy i.e., 300,000,000,000,000,000,000 m can be written as 3.0 × 100,000,000,000,000,000,000 = 3.0 × 10^20 m. Now, can you express 40,000,000,000 in the similar way? Count the number of zeros in it. It is 10. So, 40,000,000,000 = 4.0 × 10^10. Mass of the Earth = 5,976,000,000,000,000,000,000,000 kg = 5.976 × 10^24 kg. Do you agree with the fact, that the number when written in the standard form is much easier to read, understand and compare than when the number is written with 25 digits? Now, Mass of Uranus = 86,800,000,000,000,000,000,000,000 kg = 8.68 × 10^25 kg. Simply by comparing the powers of 10 in the above two, you can tell that the mass of Uranus is greater than that of the Earth. The distance between Sun and Saturn is 1,433,500,000,000 m or 1.4335 × 10^12 m. The distance between Saturn and Uranus is 1,439,000,000,000 m or 1.439 × 10^12 m. The distance between Sun and Earth is 149,600,000,000 m or 1.496 × 10^11 m. Can you tell which of the three distances is smallest? The distance between Sun and Earth is smallest. [CHECKPOINT]
Example 13 asks to express the following numbers in the standard form: 5985.3, 65,950, 3,430,000, 70,040,000,000. Solution: 5985.3 = 5.9853 × 1000 = 5.9853 × 10^3. 65,950 = 6.595 × 10,000 = 6.595 × 10^4. 3,430,000 = 3.43 × 1,000,000 = 3.43 × 10^6. 70,040,000,000 = 7.004 × 10,000,000,000 = 7.004 × 10^10. A point to remember is that one less than the digit count to the left of the decimal point in a given number is the exponent of 10 in the standard form. Thus, in 70,040,000,000 there is no decimal point shown; we assume it to be at the right end. From there, the count of the places to the left is 11. The exponent of 10 in the standard form is 11 – 1 = 10. In 5985.3 there are 4 digits to the left of the decimal point and hence the exponent of 10 in the standard form is 4 – 1 = 3. Now let us solve Exercise 11.3. Question 1: Write in expanded forms. 279404 = 2 × 10^5 + 7 × 10^4 + 9 × 10^3 + 4 × 10^2 + 0 × 10^1 + 4 × 10^0. 3006194 = 3 × 10^6 + 0 × 10^5 + 0 × 10^4 + 6 × 10^3 + 1 × 10^2 + 9 × 10^1 + 4 × 10^0. 2806196 = 2 × 10^6 + 8 × 10^5 + 0 × 10^4 + 6 × 10^3 + 1 × 10^2 + 9 × 10^1 + 6 × 10^0. 120719 = 1 × 10^5 + 2 × 10^4 + 0 × 10^3 + 7 × 10^2 + 1 × 10^1 + 9 × 10^0. 20068 = 2 × 10^4 + 0 × 10^3 + 0 × 10^2 + 6 × 10^1 + 8 × 10^0. Question 2: Find the number from expanded forms. Part a: 8 × 10^4 + 6 × 10^3 + 0 × 10^2 + 4 × 10^1 + 5 × 10^0 = 86045. Part b: 4 × 10^5 + 5 × 10^3 + 3 × 10^2 + 2 × 10^0 = 405302. Part c: 3 × 10^4 + 7 × 10^2 + 5 × 10^0 = 30705. Part d: 9 × 10^5 + 2 × 10^2 + 3 × 10^1 = 900230. Question 3: Express in standard form. Part i: 5,00,00,000 = 5 × 10^7. Part ii: 70,00,000 = 7 × 10^6. Part iii: 3,18,65,00,000 = 3.1865 × 10^9. Part iv: 3,90,878 = 3.90878 × 10^5. Part v: 39087.8 = 3.90878 × 10^4. Part vi: 3908.78 = 3.90878 × 10^3. Question 4: Express numbers in statements in standard form. Part a: 384,000,000 m = 3.84 × 10^8 m. Part b: 300,000,000 m/s = 3 × 10^8 m/s. Part c: 1,27,56,000 m = 1.2756 × 10^7 m. Part d: 1,400,000,000 m = 1.4 × 10^9 m. Part e: 100,000,000,000 stars = 1 × 10^11 stars. Part f: 12,000,000,000 years = 1.2 × 10^10 years. Part g: 300,000,000,000,000,000,000 m = 3 × 10^20 m. Part h: 60,230,000,000,000,000,000,000 = 6.023 × 10^22. Part i: 1,353,000,000 cubic km = 1.353 × 10^9 cubic km. Part j: 1,027,000,000 = 1.027 × 10^9. [CHECKPOINT]
Finally, let us review what we have discussed. Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form. The following are exponential forms of some numbers: 10,000 = 10^4, read as 10 raised to 4. 243 = 3^5, 128 = 2^7. Here, 10, 3 and 2 are the bases, whereas 4, 5 and 7 are their respective exponents. We also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc. Numbers in exponential form obey certain laws, which are: For any non-zero integers a and b and whole numbers m and n, (a) a^m × a^n = a^(m+n). (b) a^m ÷ a^n = a^(m–n), m > n. (c) (a^m)^n = a^(mn). (d) a^m × b^m = (ab)^m. (e) a^m ÷ b^m = (a/b)^m. (f) a^0 = 1. (g) (–1) raised to an even number equals 1, and (–1) raised to an odd number equals –1. Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]