Operations with Surds. Grade 10 Mathematics: Surds Subtopics Navigator Introduction Meaning of Surds Simplifying Surds Operations with Surds Rationalizing Denominators Applications Cumulative Exercises Conclusion Lesson Objectives Understand what surds are and identify them in mathematical expressions. Simplify surds using factorization and properties of square roots. Perform basic operations (addition, subtraction, multiplication, division) with surds. Rationalize denominators containing surds. Apply surds to solve real-world problems. Lesson Introduction Surds are irrational numbers that cannot be simplified to remove a square root (or cube root, etc.). They are numbers left in root form to express their exact value. For example, [latex]sqrt{2}[/latex] is a surd because it cannot be simplified to a rational number. Surds are important in mathematics because they allow us to work with exact values rather than decimal approximations. Meaning of Surds A surd is an expression containing a square root (or other root) that cannot be simplified to a rational number. The most common surds involve square roots of non-perfect squares. Example 1: Identify which of the following are surds: [latex]sqrt{4}, sqrt{5}, sqrt{9}, sqrt{12}, sqrt{25}[/latex] Solution: [latex]sqrt{4} = 2[/latex] (rational) - not a surd [latex]sqrt{5}[/latex] (irrational) - surd [latex]sqrt{9} = 3[/latex] (rational) - not a surd [latex]sqrt{12}[/latex] (irrational) - surd [latex]sqrt{25} = 5[/latex] (rational) - not a surd Exercises (Meaning of Surds) Which of the following are surds: [latex]sqrt{16}, sqrt{7}, sqrt{36}, sqrt{18}[/latex]? Explain why [latex]sqrt{50}[/latex] is considered a surd. Is [latex]sqrt{0.25}[/latex] a surd? Justify your answer. Simplifying Surds To simplify a surd, we express it in the form [latex]asqrt{b}[/latex] where b has no perfect square factors (other than 1). We do this by factorizing the number under the root and extracting perfect squares. Example 2: Simplify [latex]sqrt{72}[/latex] Solution: Factorize 72: [latex]72 = 36 times 2 = 6^2 times 2[/latex] So, [latex]sqrt{72} = sqrt{36 times 2} = sqrt{36} times sqrt{2} = 6sqrt{2}[/latex] Example 3: Simplify [latex]sqrt{128}[/latex] Solution: Factorize 128: [latex]128 = 64 times 2 = 8^2 times 2[/latex] So, [latex]sqrt{128} = sqrt{64 times 2} = sqrt{64} times sqrt{2} = 8sqrt{2}[/latex] Exercises (Simplifying Surds) Simplify [latex]sqrt{50}[/latex] Simplify [latex]sqrt{98}[/latex] Simplify [latex]sqrt{200}[/latex] Operations with Surds We can perform addition, subtraction, multiplication, and division with surds following specific rules: Addition/Subtraction: Only like surds can be added/subtracted (same radicand) Multiplication: [latex]sqrt{a} times sqrt{b} = sqrt{a times b}[/latex] Division: [latex]frac{sqrt{a}}{sqrt{b}} = sqrt{frac{a}{b}}[/latex] Example 4: Simplify [latex]3sqrt{5} + 2sqrt{5} - sqrt{5}[/latex] Solution: These are like surds, so we can combine them: [latex]3sqrt{5} + 2sqrt{5} - sqrt{5} = (3 + 2 - 1)sqrt{5} = 4sqrt{5}[/latex] Example 5: Simplify [latex]sqrt{3} times sqrt{12}[/latex] Solution: [latex]sqrt{3} times sqrt{12} = sqrt{3 times 12} = sqrt{36} = 6[/latex] Exercises (Operations with Surds) Simplify [latex]2sqrt{7} + 5sqrt{7}[/latex] Simplify [latex]sqrt{8} times sqrt{2}[/latex] Simplify [latex]4sqrt{3} - sqrt{12}[/latex] Rationalizing Denominators Rationalizing the denominator means eliminating surds from the denominator of a fraction. We do this by multiplying the numerator and denominator by the conjugate or an appropriate surd. Example 6: Rationalize the denominator of [latex]frac{3}{sqrt{5}}[/latex] Solution: Multiply numerator and denominator by [latex]sqrt{5}[/latex]: [latex]frac{3}{sqrt{5}} times frac{sqrt{5}}{sqrt{5}} = frac{3sqrt{5}}{5}[/latex] Example 7: Rationalize the denominator of [latex]frac{2}{3+sqrt{2}}[/latex] Solution: Multiply numerator and denominator by the conjugate [latex]3-sqrt{2}[/latex]: [latex]frac{2}{3+sqrt{2}} times frac{3-sqrt{2}}{3-sqrt{2}} = frac{2(3-sqrt{2})}{9-2} = frac{6-2sqrt{2}}{7}[/latex] Rationalizing Denominators Rationalize the denominator of [latex]frac{5}{sqrt{3}}[/latex] Rationalize the denominator of [latex]frac{4}{2-sqrt{5}}[/latex] Rationalize the denominator of [latex]frac{sqrt{2}}{sqrt{8}}[/latex] Applications Surds appear in various real-world contexts, particularly in geometry, physics, and engineering where exact values are required. Example 8: Find the exact length of the diagonal of a square with side length 5 cm. Solution: Using Pythagoras' theorem: [latex]d^2 = 5^2 + 5^2 = 50[/latex] So, [latex]d = sqrt{50} = 5sqrt{2}[/latex] cm (exact value) Example 9: A right triangle has legs of length [latex]sqrt{8}[/latex] cm and [latex]sqrt{18}[/latex] cm. Find the exact length of the hypotenuse. Solution: Simplify the surds first: [latex]sqrt{8} = 2sqrt{2}[/latex], [latex]sqrt{18} = 3sqrt{2}[/latex] Using Pythagoras' theorem: [latex]h^2 = (2sqrt{2})^2 + (3sqrt{2})^2 = 8 + 18 = 26[/latex] So, [latex]h = sqrt{26}[/latex] cm (exact value) Exercises (Applications) Find the exact perimeter of a square with area 32 cm². A ladder is leaning against a wall. The foot of the ladder is 2 m from the wall, and the top reaches [latex]sqrt{12}[/latex] m up the wall. What is the exact length of the ladder? The diagonal of a rectangle is [latex]sqrt{50}[/latex] cm, and its width is [latex]sqrt{18}[/latex] cm. Find the exact length of the rectangle. Cumulative Exercises Simplify [latex]sqrt{75} + sqrt{48} - sqrt{27}[/latex] Rationalize the denominator of [latex]frac{3}{sqrt{7} - sqrt{2}}[/latex] Simplify [latex](2sqrt{3} + sqrt{5})(2sqrt{3} - sqrt{5})[/latex] Find the exact value of [latex]frac{sqrt{20} + sqrt{45}}{sqrt{5}}[/latex] Simplify [latex]sqrt{8} times sqrt{18} div sqrt{2}[/latex] Rationalize the denominator of [latex]frac{6}{3sqrt{2} + 2sqrt{3}}[/latex] Simplify [latex](sqrt{3} + 1)^2 - (sqrt{3} - 1)^2[/latex] Find the exact perimeter of a rectangle with length [latex]sqrt{12}[/latex] cm and width [latex]sqrt{27}[/latex] cm Simplify [latex]frac{sqrt{50} - sqrt{18}}{sqrt{2}}[/latex] If [latex]x = sqrt{5} + 2[/latex] and [latex]y = sqrt{5} - 2[/latex], find the value of [latex]x^2 + y^2[/latex] Show/Hide Answers Problem: Simplify [latex]sqrt{75} + sqrt{48} - sqrt{27}[/latex] Step 1: Simplify each surd individually [latex]sqrt{75} = sqrt{25 times 3} = 5sqrt{3}[/latex] [latex]sqrt{48} = sqrt{16 times 3} = 4sqrt{3}[/latex] [latex]sqrt{27} = sqrt{9 times 3} = 3sqrt{3}[/latex] Step 2: Combine like terms [latex]5sqrt{3} + 4sqrt{3} - 3sqrt{3} = (5 + 4 - 3)sqrt{3} = 6sqrt{3}[/latex] Answer: [latex]6sqrt{3}[/latex] Problem: Rationalize the denominator of [latex]frac{3}{sqrt{7} - sqrt{2}}[/latex] Step 1: Multiply numerator and denominator by the conjugate [latex]sqrt{7} + sqrt{2}[/latex] [latex]frac{3}{sqrt{7} - sqrt{2}} times frac{sqrt{7} + sqrt{2}}{sqrt{7} + sqrt{2}} = frac{3(sqrt{7} + sqrt{2})}{(sqrt{7})^2 - (sqrt{2})^2}[/latex] Step 2: Simplify denominator [latex](sqrt{7})^2 - (sqrt{2})^2 = 7 - 2 = 5[/latex] Step 3: Final expression [latex]frac{3(sqrt{7} + sqrt{2})}{5} = frac{3sqrt{7} + 3sqrt{2}}{5}[/latex] Answer: [latex]frac{3sqrt{7} + 3sqrt{2}}{5}[/latex] Problem: Simplify [latex](2sqrt{3} + sqrt{5})(2sqrt{3} - sqrt{5})[/latex] Step 1: Recognize this as difference of squares: [latex](a+b)(a-b) = a^2 - b^2[/latex] [latex](2sqrt{3})^2 - (sqrt{5})^2[/latex] Step 2: Calculate each term [latex](2sqrt{3})^2 = 4 times 3 = 12[/latex] [latex](sqrt{5})^2 = 5[/latex] Step 3: Subtract [latex]12 - 5 = 7[/latex] Answer: [latex]7[/latex] Problem: Find the exact value of [latex]frac{sqrt{20} + sqrt{45}}{sqrt{5}}[/latex] Step 1: Simplify each surd in numerator [latex]sqrt{20} = sqrt{4 times 5} = 2sqrt{5}[/latex] [latex]sqrt{45} = sqrt{9 times 5} = 3sqrt{5}[/latex] Step 2: Substitute back into expression [latex]frac{2sqrt{5} + 3sqrt{5}}{sqrt{5}} = frac{5sqrt{5}}{sqrt{5}}[/latex] Step 3: Simplify [latex]frac{5sqrt{5}}{sqrt{5}} = 5[/latex] Answer: [latex]5[/latex] Problem: Simplify [latex]sqrt{8} times sqrt{18} div sqrt{2}[/latex] Step 1: Combine under single square root [latex]sqrt{frac{8 times 18}{2}}[/latex] Step 2: Simplify inside the root [latex]frac{8 times 18}{2} = 8 times 9 = 72[/latex] Step 3: Simplify the surd [latex]sqrt{72} = sqrt{36 times 2} = 6sqrt{2}[/latex] Answer: [latex]6sqrt{2}[/latex] Problem: Rationalize the denominator of [latex]frac{6}{3sqrt{2} + 2sqrt{3}}[/latex] Step 1: Multiply numerator and denominator by the conjugate [latex]3sqrt{2} - 2sqrt{3}[/latex] [latex]frac{6}{3sqrt{2} + 2sqrt{3}} times frac{3sqrt{2} - 2sqrt{3}}{3sqrt{2} - 2sqrt{3}}[/latex] Step 2: Simplify denominator using difference of squares [latex](3sqrt{2})^2 - (2sqrt{3})^2 = (9 times 2) - (4 times 3) = 18 - 12 = 6[/latex] Step 3: Final expression [latex]frac{6(3sqrt{2} - 2sqrt{3})}{6} = 3sqrt{2} - 2sqrt{3}[/latex] Answer: [latex]3sqrt{2} - 2sqrt{3}[/latex] Problem: Simplify [latex](sqrt{3} + 1)^2 - (sqrt{3} - 1)^2[/latex] Step 1: Expand each square [latex](sqrt{3} + 1)^2 = 3 + 2sqrt{3} + 1 = 4 + 2sqrt{3}[/latex] [latex](sqrt{3} - 1)^2 = 3 - 2sqrt{3} + 1 = 4 - 2sqrt{3}[/latex] Step 2: Subtract the expressions [latex](4 + 2sqrt{3}) - (4 - 2sqrt{3}) = 4 + 2sqrt{3} - 4 + 2sqrt{3} = 4sqrt{3}[/latex] Answer: [latex]4sqrt{3}[/latex] Problem: Find the exact perimeter of a rectangle with length [latex]sqrt{12}[/latex] cm and width [latex]sqrt{27}[/latex] cm Step 1: Simplify the surds [latex]sqrt{12} = sqrt{4 times 3} = 2sqrt{3}[/latex] cm [latex]sqrt{27} = sqrt{9 times 3} = 3sqrt{3}[/latex] cm Step 2: Calculate perimeter: [latex]P = 2(l + w)[/latex] [latex]P = 2(2sqrt{3} + 3sqrt{3}) = 2(5sqrt{3}) = 10sqrt{3}[/latex] cm Answer: [latex]10sqrt{3}[/latex] cm Problem: Simplify [latex]frac{sqrt{50} - sqrt{18}}{sqrt{2}}[/latex] Step 1: Simplify numerator surds [latex]sqrt{50} = sqrt{25 times 2} = 5sqrt{2}[/latex] [latex]sqrt{18} = sqrt{9 times 2} = 3sqrt{2}[/latex] Step 2: Substitute back [latex]frac{5sqrt{2} - 3sqrt{2}}{sqrt{2}} = frac{2sqrt{2}}{sqrt{2}}[/latex] Step 3: Simplify [latex]frac{2sqrt{2}}{sqrt{2}} = 2[/latex] Answer: [latex]2[/latex] Problem: If [latex]x = sqrt{5} + 2[/latex] and [latex]y = sqrt{5} - 2[/latex], find the value of [latex]x^2 + y^2[/latex] Step 1: Calculate [latex]x^2[/latex] [latex]x^2 = (sqrt{5} + 2)^2 = 5 + 4sqrt{5} + 4 = 9 + 4sqrt{5}[/latex] Step 2: Calculate [latex]y^2[/latex] [latex]y^2 = (sqrt{5} - 2)^2 = 5 - 4sqrt{5} + 4 = 9 - 4sqrt{5}[/latex] Step 3: Add [latex]x^2 + y^2[/latex] [latex](9 + 4sqrt{5}) + (9 - 4sqrt{5}) = 9 + 4sqrt{5} + 9 - 4sqrt{5} = 18[/latex] Answer: [latex]18[/latex] Conclusion/Recap Surds are irrational numbers expressed in root form, which allow us to work with exact values in mathematics. We can simplify surds by extracting perfect squares, perform operations with them following specific rules, and rationalize denominators to eliminate surds from the bottom of fractions. These skills are essential for advanced mathematics and have practical applications in geometry and other fields. Clip It! Share your ANSWER in the Chat. Indicate TITLE e.g Linear Equation 1. .....2. e.t.c
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