This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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here are some key notes on surds, focusing on what's typically examinable at the UCE level: 1. Definition of a Surd A surd is an irrational number that is expressed as a root of an integer. This means its value cannot be expressed exactly as a simple fraction. For example, sqrt(2), sqrt(3), and sqrt(5) are surds. However, sqrt(4) is not a surd because it simplifies to 2, which is a rational number. Similarly, [3]8 is not a surd because it simplifies to 2. 2. Simplifying Surds To simplify a surd, find the largest perfect square factor of the number under the square root sign. Use the property sqrt(a × b) = sqrt(a) × sqrt(b). Example: Simplify sqrt(72). Step 1: Find the largest perfect square factor of 72. The largest perfect square factor of 72 is 36 (36 × 2 = 72). Step 2: Apply the property sqrt(a × b) = sqrt(a) × sqrt(b). sqrt(72) = sqrt(36 × 2) sqrt(72) = sqrt(36) × sqrt(2) sqrt(72) = 6sqrt(2) 3. Operations with Surds Addition and Subtraction: You can only add or subtract like surds* (surds with the same number under the root sign). If surds are not alike, simplify them first to see if they become like surds. Example: Simplify 3sqrt(5) + sqrt(20). Step 1: Simplify sqrt(20). sqrt(20) = sqrt(4 × 5) = sqrt(4) × sqrt(5) = 2sqrt(5) Step 2: Add the like surds. 3sqrt(5) + 2sqrt(5) = 5sqrt(5) Multiplication: Use the property sqrt(a) × sqrt(b) = sqrt(a × b). Example: Simplify sqrt(6) × sqrt(3). Step 1: Multiply the numbers under the root sign. sqrt(6) × sqrt(3) = sqrt(6 × 3) sqrt(6) × sqrt(3) = sqrt(18) Step 2: Simplify the resulting surd. sqrt(18) = sqrt(9 × 2) = sqrt(9) × sqrt(2) = 3sqrt(2) Division: Use the property sqrt(a)sqrt(b) = sqrt((a)/(b)). Example: Simplify sqrt(48)sqrt(3). Step 1: Divide the numbers under the root sign. sqrt(48)sqrt(3) = sqrt((48)/(3)) sqrt(48)sqrt(3) = sqrt(16) Step 2: Evaluate the square root. sqrt(16) = 4 4. Rationalizing the Denominator Rationalizing the denominator means removing any surds from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a suitable term. Single Term Denominator: If the denominator is of the form sqrt(a), multiply the numerator and denominator by sqrt(a). Example: Rationalize (5)/(sqrt(7)). Step 1: Multiply numerator and denominator by sqrt(7). (5)/(sqrt(7)) × sqrt(7)sqrt(7) Step 2: Perform the multiplication. 5sqrt(7)7 = 5sqrt(7)7 Binomial Denominator: If the denominator is of the form a + sqrt(b) or sqrt(a) + sqrt(b), multiply the numerator and denominator by its conjugate*. The conjugate of a + sqrt(b) is a - sqrt(b), and vice versa. This uses the difference of squares formula: (x+y)(x-y) = x^2 - y^2. Example: Rationalize (1)/(3 - sqrt(2)). Step 1: Identify the conjugate of the denominator. The conjugate of 3 - sqrt(2) is 3 + sqrt(2). Step 2: Multiply numerator and denominator by the conjugate. (1)/(3 - sqrt(2)) × 3 + sqrt(2)3 + sqrt(2) Step 3: Perform the multiplication. 1 × (3 + sqrt(2))(3 - sqrt(2))(3 + sqrt(2)) 3 + sqrt(2)3^2 - (sqrt(2))^2 3 + sqrt(2)9 - 2 3 + sqrt(2)7 = 3 + sqrt(2)7 Got more? Send 'em!