Indices & Standard Form.

Indices & Standard form. Grade 10 Mathematics: Indices and Standard Form Subtopics Navigator Introduction to Indices Laws of Indices Negative Indices Fractional Indices Standard Form Operations with Standard Form Cumulative Exercises Conclusion Lesson Objectives Understand the concept of indices and their notation Apply the laws of indices to simplify expressions Work with negative and fractional indices Convert numbers to and from standard form Perform calculations using numbers in standard form Apply indices and standard form to solve real-world problems Lesson Introduction Indices (also called exponents or powers) are a fundamental concept in mathematics that allow us to express repeated multiplication in a compact form. Standard form (scientific notation) helps us work with very large or very small numbers efficiently. These concepts are essential in science, engineering, economics, and many other fields. Index Notation: When a number is multiplied by itself multiple times, we can write it using index notation: [latex]a^n = a times a times a times cdots times a quad text{(n times)}[/latex] Where: a is the base n is the index, exponent, or power The whole expression [latex]a^n[/latex] is called a power Laws of Indices The laws of indices are rules that help us manipulate expressions with exponents efficiently and correctly: Law Rule Example Multiplication Law [latex]a^m times a^n = a^{m+n}[/latex] [latex]2^3 times 2^4 = 2^{3+4} = 2^7[/latex] Division Law [latex]a^m div a^n = a^{m-n}[/latex] [latex]5^6 div 5^2 = 5^{6-2} = 5^4[/latex] Power of a Power [latex](a^m)^n = a^{m times n}[/latex] [latex](3^2)^3 = 3^{2 times 3} = 3^6[/latex] Power of a Product [latex](ab)^n = a^n b^n[/latex] [latex](2 times 3)^4 = 2^4 times 3^4[/latex] Power of a Quotient [latex]left(frac{a}{b}right)^n = frac{a^n}{b^n}[/latex] [latex]left(frac{4}{5}right)^3 = frac{4^3}{5^3}[/latex] Zero Index [latex]a^0 = 1[/latex] (where a ≠ 0) [latex]7^0 = 1[/latex] Example 1: Applying Multiple Laws Simplify: [latex]frac{(2^3 times 3^2)^2}{2^4 times 3^3}[/latex] Solution: [latex]= frac{2^{3 times 2} times 3^{2 times 2}}{2^4 times 3^3} = frac{2^6 times 3^4}{2^4 times 3^3} = 2^{6-4} times 3^{4-3} = 2^2 times 3^1 = 4 times 3 = 12[/latex] Exercises (Laws of Indices) Simplify: [latex]5^3 times 5^4[/latex] Simplify: [latex]frac{7^8}{7^5}[/latex] Simplify: [latex](2^3)^4[/latex] Simplify: [latex](4 times 5)^2[/latex] Simplify: [latex]left(frac{3}{4}right)^5[/latex] Negative Indices A negative index indicates the reciprocal of the base raised to the positive power: Negative Index Rule: [latex]a^{-n} = frac{1}{a^n} quad text{and} quad frac{1}{a^{-n}} = a^n[/latex] Example 2: Working with Negative Indices Simplify: [latex]2^{-3} times 3^{-2}[/latex] Solution: [latex]= frac{1}{2^3} times frac{1}{3^2} = frac{1}{8} times frac{1}{9} = frac{1}{72}[/latex] Example 3: More Complex Negative Indices Simplify: [latex]frac{5^{-2}}{5^{-4}}[/latex] Solution: [latex]= 5^{-2 - (-4)} = 5^{-2 + 4} = 5^2 = 25[/latex] OR [latex]= frac{frac{1}{5^2}}{frac{1}{5^4}} = frac{1}{5^2} times frac{5^4}{1} = 5^{4-2} = 5^2 = 25[/latex] Exercises (Negative Indices) Evaluate: [latex]3^{-2}[/latex] Evaluate: [latex]2^{-4}[/latex] Simplify: [latex]frac{5^{-2}}{5^{-4}}[/latex] Simplify: [latex]2^{-3} times 2^5[/latex] Simplify: [latex]frac{4^{-2} times 4^3}{4^{-1}}[/latex] Fractional Indices A fractional index represents a root: Fractional Index Rules: [latex]a^{frac{1}{n}} = sqrt[n]{a} quad text{and} quad a^{frac{m}{n}} = (sqrt[n]{a})^m = sqrt[n]{a^m}[/latex] Example 4: Fractional Indices Evaluate: [latex]8^{frac{2}{3}}[/latex] Solution: [latex]= (sqrt[3]{8})^2 = 2^2 = 4 quad text{or} quad sqrt[3]{8^2} = sqrt[3]{64} = 4[/latex] Example 5: More Fractional Indices Evaluate: [latex]16^{frac{3}{4}}[/latex] Solution: [latex]= (sqrt[4]{16})^3 = 2^3 = 8 quad text{or} quad sqrt[4]{16^3} = sqrt[4]{4096} = 8[/latex] Exercises (Fractional Indices) Evaluate: [latex]16^{frac{1}{2}}[/latex] Evaluate: [latex]27^{frac{2}{3}}[/latex] Evaluate: [latex]32^{frac{3}{5}}[/latex] Evaluate: [latex]81^{frac{3}{4}}[/latex] Simplify: [latex](8^{frac{1}{3}})^2[/latex] Standard Form Standard form (also called scientific notation) is a way to write very large or very small numbers conveniently: Standard Form Definition: A number in standard form is written as: [latex]a times 10^n[/latex] Where: 1 ≤ a < 10 (a is between 1 and 10, not including 10) n is an integer (positive, negative, or zero) Example 6: Converting to Standard Form Large numbers: 5,600,000 = [latex]5.6 times 10^6[/latex] 123,000,000 = [latex]1.23 times 10^8[/latex] Small numbers: 0.000078 = [latex]7.8 times 10^{-5}[/latex] 0.000000432 = [latex]4.32 times 10^{-7}[/latex] Example 7: Converting from Standard Form [latex]3.2 times 10^4 = 32,000[/latex] [latex]4.5 times 10^{-3} = 0.0045[/latex] [latex]7.89 times 10^6 = 7,890,000[/latex] [latex]2.1 times 10^{-5} = 0.000021[/latex] Exercises (Standard Form) Write 450,000 in standard form Write 0.00325 in standard form Convert [latex]7.8 times 10^5[/latex] to ordinary form Convert [latex]2.1 times 10^{-4}[/latex] to ordinary form Write the distance to the sun (149,600,000 km) in standard form Operations with Standard Form When performing operations with numbers in standard form, we handle the coefficients and powers of 10 separately: Example 8: Multiplication Multiply: [latex](3 times 10^4) times (2 times 10^5)[/latex] Solution: [latex]= (3 times 2) times (10^4 times 10^5) = 6 times 10^{4+5} = 6 times 10^9[/latex] Example 9: Division Divide: [latex](8 times 10^7) div (2 times 10^3)[/latex] Solution: [latex]= (8 div 2) times (10^7 div 10^3) = 4 times 10^{7-3} = 4 times 10^4[/latex] Example 10: Addition (Same Power) Add: [latex](4.5 times 10^3) + (2.3 times 10^3)[/latex] Solution: [latex]= (4.5 + 2.3) times 10^3 = 6.8 times 10^3[/latex] Example 11: Addition (Different Powers) Add: [latex](3.2 times 10^4) + (5.1 times 10^3)[/latex] Solution: Convert to same power: [latex]3.2 times 10^4 = 32 times 10^3[/latex] [latex]= (32 times 10^3) + (5.1 times 10^3) = (32 + 5.1) times 10^3 = 37.1 times 10^3 = 3.71 times 10^4[/latex] Exercises (Operations with Standard Form) Multiply: [latex](2 times 10^3) times (3 times 10^4)[/latex] Divide: [latex](9 times 10^6) div (3 times 10^2)[/latex] Add: [latex](4.2 times 10^5) + (3.8 times 10^5)[/latex] Subtract: [latex](7.5 times 10^3) - (2.3 times 10^3)[/latex] Add: [latex](6.3 times 10^4) + (4.2 times 10^3)[/latex] Cumulative Exercises Simplify: [latex]2^3 times 2^5[/latex] Simplify: [latex]frac{5^7}{5^4}[/latex] Simplify: [latex](3^2)^4[/latex] Evaluate: [latex]4^{-2}[/latex] Evaluate: [latex]16^{frac{1}{2}}[/latex] Evaluate: [latex]27^{frac{2}{3}}[/latex] Write 67,000,000 in standard form Write 0.000045 in standard form Convert [latex]3.4 times 10^6[/latex] to ordinary form Convert [latex]7.2 times 10^{-5}[/latex] to ordinary form Multiply: [latex](4 times 10^3) times (5 times 10^4)[/latex] Divide: [latex](8 times 10^7) div (2 times 10^2)[/latex] Add: [latex](3.5 times 10^4) + (2.5 times 10^4)[/latex] The mass of Earth is approximately [latex]5.97 times 10^{24}[/latex] kg. The mass of the Moon is approximately [latex]7.35 times 10^{22}[/latex] kg. How many times heavier is Earth than the Moon? A computer can perform [latex]4 times 10^9[/latex] operations per second. How many operations can it perform in one hour? Show/Hide Answers Problem: Simplify: [latex]2^3 times 2^5[/latex] Solution: [latex]2^{3+5} = 2^8 = 256[/latex] Answer: 256 Problem: Simplify: [latex]frac{5^7}{5^4}[/latex] Solution: [latex]5^{7-4} = 5^3 = 125[/latex] Answer: 125 Problem: Simplify: [latex](3^2)^4[/latex] Solution: [latex]3^{2 times 4} = 3^8 = 6561[/latex] Answer: 6561 Problem: Evaluate: [latex]4^{-2}[/latex] Solution: [latex]frac{1}{4^2} = frac{1}{16}[/latex] Answer: [latex]frac{1}{16}[/latex] Problem: Evaluate: [latex]16^{frac{1}{2}}[/latex] Solution: [latex]sqrt{16} = 4[/latex] Answer: 4 Problem: Evaluate: [latex]27^{frac{2}{3}}[/latex] Solution: [latex](sqrt[3]{27})^2 = 3^2 = 9[/latex] Answer: 9 Problem: Write 67,000,000 in standard form Solution: [latex]6.7 times 10^7[/latex] Answer: [latex]6.7 times 10^7[/latex] Problem: Write 0.000045 in standard form Solution: [latex]4.5 times 10^{-5}[/latex] Answer: [latex]4.5 times 10^{-5}[/latex] Problem: Convert [latex]3.4 times 10^6[/latex] to ordinary form Solution: 3,400,000 Answer: 3,400,000 Problem: Convert [latex]7.2 times 10^{-5}[/latex] to ordinary form Solution: 0.000072 Answer: 0.000072 Problem: Multiply: [latex](4 times 10^3) times (5 times 10^4)[/latex] Solution: [latex](4 times 5) times 10^{3+4} = 20 times 10^7 = 2.0 times 10^8[/latex] Answer: [latex]2.0 times 10^8[/latex] Problem: Divide: [latex](8 times 10^7) div (2 times 10^2)[/latex] Solution: [latex](8 div 2) times 10^{7-2} = 4 times 10^5[/latex] Answer: [latex]4 times 10^5[/latex] Problem: Add: [latex](3.5 times 10^4) + (2.5 times 10^4)[/latex] Solution: [latex](3.5 + 2.5) times 10^4 = 6.0 times 10^4[/latex] Answer: [latex]6.0 times 10^4[/latex] Problem: How many times heavier is Earth than the Moon? Solution: [latex]frac{5.97 times 10^{24}}{7.35 times 10^{22}} = frac{5.97}{7.35} times 10^{24-22} = 0.812 times 10^2 = 81.2[/latex] Answer: Earth is about 81.2 times heavier than the Moon Problem: Operations in one hour Solution: 1 hour = 3600 seconds [latex]4 times 10^9 times 3.6 times 10^3 = 14.4 times 10^{12} = 1.44 times 10^{13}[/latex] Answer: [latex]1.44 times 10^{13}[/latex] operations Conclusion/Recap In this lesson, we've covered the fundamental concepts of indices and standard form. We learned the laws of indices for multiplication, division, and powers, worked with negative and fractional indices, and mastered standard form for representing very large and very small numbers. These skills are essential for advanced mathematics and have practical applications in science, engineering, and everyday life. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c