Real Number system and surds class 8 Vedanta publications solutions || optional math || solutions

Long Questions : 10. Simplify: (a) 5/125-645 + 2√5 (b) (d) 20 + 4sqrt(80 - 2sqrt(180); 40) + 3 * root(625, 3) - root(320, 3) (c) 4√90-6√160 + √245 - 5/125 11. Arrange the following surds in ascending order : (a) 5, √2 and √7 (b) √7, 15 and √3 (c) √5, 11 and 2/3 12. Rationalize the denominators and simplify : (a) (7 + sqrt(3))/(7 - sqrt(3)) b (2 + sqrt(5))/(2 - sqrt(5)) (c) (sqrt(5) + sqrt(2))/(sqrt(5) - sqrt(2)) 383×2=6 3.6 Rationalization of Surds When the product of two surds is a rational number, each of them is said to be the rationalizing factor of the other. Example: √3+√2 is rationalizing factor of √3-√2. Then, (√3+√2) x (√3-√2)=3-2 = 1. The process from which a surd is changed to a rational number by multiplying it with a suitable factor is called the rationalization of the surd.. The following are some basic steps for rationalization. (i) Multiply the given surd by a simplest rationalizing factor. Example: 4√2 × √2 = 4 × 2 = 8 Here √2 is a simplest rationalizing factor. (ii) Multiply a binomial surd by its conjugate. Example: √3+√2, its conjugate is √3-√2 Now, (√3 + √2) x (√3-√2) = (√3)² - (√2)² = 3-2 = 1 Here, √3-√2 is a rationalizing factor of √3 + √2. (iii) When a surd is in the form of quotient, multiply both the numerator and denominator by conjugate of the denominator to make denominator a rational number. 2+√5 Example: 5-2 44 (multiplying numerator and denominator by √5 + 2) 2+55+2 √5-25 5+2 = = (√5+2)²(5)+2.5.2+2 (5)-2 5-4 5+ 4/5 + 4 = 9+4/5 1 n is Example: (i) √7+ √5 and √7-√5 are conjugate surds. (ii) 2√3-2√2 and 2√3 2sqrt(2) are conjugate surds. Laws of Surd Let a and b be two positive real numbers and m, n and p be integers. Then. he surd. er of the of unlike (i) a = a (ii) a. V = Vab na (iv) sqrt[n] ( sqrt[n] a = root(a, mn) = root(root(a, m), n) (v) mn 3.5 Operation of Surds e.g. 3 = 3 e.g. sqrt(5) * sqrt(3) = sqrt(5 * 3) = sqrt(15) e.g. (root(4, 3))/(root(2, 3)) = root(4/2, 3) = root(2, 3) e.g. == 1/4 e.g. root(2, 3) = root(2 ^ 4, 3 * 4) = root(16, 12) = 2 ^ (4 * (- 1/12)) = 2 ^ (1/3) (i) Addition and Subtraction of Surds Two or more like surds can be added or subtracted. To add or subtract surds, follow the given ways: (a) Express all the like surds into simplest form. (b) Add or subtract the coefficients of like terms keeping the irrational factor same. Examples: (i) overline 7 + 3sqrt(7) = (1 + 3) * sqrt(7) = 4sqrt(7) (ii) 4√3-2√3() (iii) √18 + √50 = √9x2+√25 × 2 = 3sqrt(2) + 5sqrt(2) = (3 + 5) * sqrt(2) = 8sqrt(2) (iv) overline 2 + sqrt(8) - sqrt(18) = 4√2 + 2√2-3√2 =(4+2-3)√2 = 3√2 ed surd. (ii) Multiplication and Division of Surds: Two or more surds of same order c an be multiplied. Example: Ivedantal Erru(11) 5x25 =45×25-5-5 (iii) sqrt(7) * 4sqrt(7) sqrt * 7 Wor 12x7-84 Note: Va Vab Beaample 17 Add: 4/7+8/7 Solution: Here. 4/7+/7 A surd can be divided by another surd of the same order. Example: (1) 1sqrt(7) / 2 * sqrt(7) 4 sqrt prime 2sqrt(7) = 2 = ( t + mv Example 2. Subtract: 8/5- Solution: Here, 8/5-5/5 (H) 3/72+V6 3/72 72 3V9 (8-55 3/8=3×2=6 sample 3: Multiplys 3.6 Rationalization of Surds (i) sqrt(3) * sqrt(2) = sqrt(3 * 2) = sqrt(6) al Mathematics Book 8 43