Find the value of a and b in the following if 1 4 + 15 = a + b 15

We are given to find the value of a and b in the following if 1 4 + 15 = a + b 15 . We rationalise the denominator when the given fraction contains a radical term or a surd in the denominator. If the denominator of a mathematical expression with two terms includes radical, then we need to multiply both the numerator and denominator by the conjugate of the denominator. This method is called rationalization. The expression that is multiplied with an irrational expression to obtain a rational number is called the "rationalising factor". Thus, we have, 1 4 + 15 = a + b 15 ⇒ 1 × ( 4 - 15 ) 4 + 15 × ( 4 - 15 ) = a + b 15 ⇒ 4 - 15 4 2 - 15 2 = a + b 15 ∵ ( a + b ) ( a - b ) = a 2 - b 2 ⇒ 4 - 15 16 - 15 = a + b 15 ⇒ 4 - 15 1 = a + b 15 ⇒ 4 - 15 = a + b 15 Now, comparing the rational and irrational parts of both sides of the equation, we obtain, a = 4 and b = - 1 .

Find the value of a and b in the following if 1 3 + 7 = a + b 7

We are given to find the value of a and b in the following if 1 3 + 7 = a + b 7 . We rationalise the denominator when the given fraction contains a radical term or a surd in the denominator. If the denominator of a mathematical expression with two terms includes radical, then we need to multiply both the numerator and denominator by the conjugate of the denominator. This method is called rationalization. The expression that is multiplied with an irrational expression to obtain a rational number is called the "rationalising factor". Thus, we have, 1 3 + 7 = a + b 7 ⇒ 1 × ( 3 - 7 ) 3 + 7 × ( 3 - 7 ) = a + b 7 ⇒ 3 - 7 3 2 - 7 2 = a + b 7 ∵ ( a + b ) ( a - b ) = a 2 - b 2 ⇒ 3 - 7 9 - 7 = a + b 7 ⇒ 3 - 7 2 = a + b 7 ⇒ 3 2 - 1 2 7 = a + b 7 Now, comparing the rational and irrational parts of both sides of the equation, we obtain, a = 3 2 and b = - 1 2 .

Find the value of a and b in the following if 4 3 2 - 3 = a + b 3

We are given to find the value of a and b in the following if 4 3 2 - 3 = a + b 3 . We rationalise the denominator when the given fraction contains a radical term or a surd in the denominator. If the denominator of a mathematical expression with two terms includes radical, then we need to multiply both the numerator and denominator by the conjugate of the denominator. This method is called rationalization. The expression that is multiplied with an irrational expression to obtain a rational number is called the "rationalising factor". Thus, we have, 4 3 2 - 3 = a + b 3 ⇒ 4 3 × ( 2 + 3 ) 2 - 3 × ( 2 + 3 ) = a + b 3 ⇒ 4 3 × 2 + 4 3 × 3 2 2 - 3 2 = a + b 3 ⇒ 8 3 + 12 4 - 3 = a + b 3 ∵ ( a + b ) ( a - b ) = a 2 - b 2 ⇒ 12 + 8 3 1 = a + b 3 ⇒ 12 + 8 3 = a + b 3 Now, comparing the rational and irrational parts of both sides of the equation, we obtain, a = 12 and b = 8 .

Find the value of a and b in the following if 5 3 + 3 2 3 - 3 = a + b 3

We are given to find the value of a and b in the following if 5 3 + 3 2 3 - 3 = a + b 3 . We rationalise the denominator when the given fraction contains a radical term or a surd in the denominator. If the denominator of a mathematical expression with two terms includes radical, then we need to multiply both the numerator and denominator by the conjugate of the denominator. This method is called rationalization. The expression that is multiplied with an irrational expression to obtain a rational number is called the "rationalising factor". Thus, we have, 5 3 + 3 2 3 - 3 = a + b 3 ⇒ 5 3 + 3 × ( 2 3 + 3 ) 2 3 - 3 × ( 2 3 + 3 ) = a + b 3 ⇒ 5 3 × ( 2 3 + 3 ) + 3 × ( 2 3 + 3 ) ( 2 3 - 3 ) × ( 2 3 + 3 ) = a + b 3 ⇒ 5 3 × 2 3 + 3 × 5 3 + 3 × 2 3 + 3 × 9 2 3 2 - 3 2 = a + b 3 ∵ ( a + b ) ( a - b ) = a 2 - b 2 ⇒ 30 + 15 3 + 6 3 + 27 12 - 9 = a + b 3 ⇒ 57 + 21 3 3 = a + b 3 ⇒ 3 19 + 7 3 3 = a + b 3 ⇒ 19 + 7 3 = a + b 3 Now, comparing the rational and irrational parts of both sides of the equation, we obtain, a = 19 and b = 7 .

Find the value of a and b in the following if 5 + 5 9 - 4 5 = a + b 5

We are given to find the value of a and b in the following if 5 + 5 9 - 4 5 = a + b 5 . We rationalise the denominator when the given fraction contains a radical term or a surd in the denominator. If the denominator of a mathematical expression with two terms includes radical, then we need to multiply both the numerator and denominator by the conjugate of the denominator. This method is called rationalization. The expression that is multiplied with an irrational expression to obtain a rational number is called the "rationalising factor". Thus, we have, 5 + 5 9 - 4 5 = a + b 5 ⇒ 5 + 5 × ( 9 + 4 5 ) 9 - 4 5 × ( 9 + 4 5 ) = a + b 5 ⇒ 5 × ( 9 + 4 5 ) + 5 × ( 9 + 4 5 ) 9 2 - 4 5 2 = a + b 5 ∵ ( a + b ) ( a - b ) = a 2 - b 2 ⇒ 45 + 20 5 + 9 5 + 20 81 - 80 = a + b 5 ⇒ 65 + 29 5 1 = a + b 5 ⇒ 65 + 29 5 = a + b 5 Now, comparing the rational and irrational parts of both sides of the equation, we obtain, a = 65 and b = 29 .

Find the value of a and b in the following if 6 + 3 7 - 4 3 = a + b 3 .

We are given to find the value of a and b in the following if 6 + 3 7 - 4 3 = a + b 3 . We rationalise the denominator when the given fraction contains a radical term or a surd in the denominator. If the denominator of a mathematical expression with two terms includes radical, then we need to multiply both the numerator and denominator by the conjugate of the denominator. This method is called rationalization. The expression that is multiplied with an irrational expression to obtain a rational number is called the "rationalising factor". Thus, we have, 6 + 3 7 - 4 3 = a + b 3 ⇒ 6 + 3 × ( 7 + 4 3 ) 7 - 4 3 × ( 7 + 4 3 ) = a + b 3 ⇒ 6 × ( 7 + 4 3 ) + 3 × ( 7 + 4 3 ) 7 2 - 4 3 2 = a + b 3 ∵ ( a + b ) ( a - b ) = a 2 - b 2 ⇒ 6 × 7 + 6 × 4 3 + 3 × 7 + 3 × 4 3 49 - 48 = a + b 3 ⇒ 42 + 24 3 + 7 3 + 12 1 = a + b 3 ⇒ 54 + 31 5 = a + b 5 Now, comparing the rational and irrational parts of both sides of the equation, we obtain, a = 54 and b = 31 .

Insert 3 rational numbers between 3 7 and 7 9 .

We are given to insert 3 rational numbers between 3 7 and 7 9 . We shall use the method of the common denominator to find the rational numbers between the given rational numbers. The LCM of the denominators of the rational numbers 3 7 and 7 9 is 63 . Now, we have, 3 7 = 3 × 9 7 × 9 = 27 63 and 7 9 = 7 × 7 9 × 7 = 49 63 . Now, any rational number whose denominator is 63 and the numerator is any number between 27 and 49 will be the rational numbers between 3 7 and 7 9 . We can choose any number between 27 and 49 as numerators. Arbitrarily, we are choosing 29 , 31 and 37 as numerators. Therefore, the three rational numbers between the given rational numbers 3 7 and 7 9 are 29 63 , 31 63 and 37 63 .