When working out some of the diagonal lengths involved in calculating the size of individual plots, Bert creates the following expression:
fraction numerator 10 plus 15 square root of 5 over denominator square root of 5 minus 2 end fraction
Rationalize the denominator and simplify this expression. What is the value of b if the answer is expressed in the format a plus b square root of 5?
To rationalize the denominator of the expression, we use the conjugate. The conjugate of the square root of 5 minus 2 is the square root of 5 plus 2.
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate:
((10 + 15 * sqrt(5)) / (sqrt(5) - 2)) * ((sqrt(5) + 2) / (sqrt(5) + 2))
This simplifies to:
((10 + 15 * sqrt(5)) * (sqrt(5) + 2)) / ((sqrt(5))^2 - (2)^2)
Simplifying further, we have:
(10 * sqrt(5) + 2 * sqrt(5) + (15 * sqrt(5))^2 + 30 * sqrt(5)) / (5 - 4)
Expanding the squares:
(10 * sqrt(5) + 2 * sqrt(5) + 225 * 5 + 30 * sqrt(5)) / 1
Combining like terms:
(10 * sqrt(5) + 2 * sqrt(5) + 1125 + 30 * sqrt(5))
Simplifying this expression, we have:
(1125 + 42 * sqrt(5))
Therefore, the value of b is 42.
To rationalize the denominator of the given expression, we multiply both the numerator and the denominator by the conjugate of the denominator.
The conjugate of the denominator, square root of 5 minus 2, would be square root of 5 plus 2.
So, we multiply the expression by square root of 5 plus 2 over square root of 5 plus 2:
((10 + 15 square root of 5) / (square root of 5 - 2)) * ((square root of 5 + 2) / (square root of 5 + 2))
Next, we simplify the numerator and denominator:
Numerator: (10 + 15 square root of 5)(square root of 5 + 2) = 10 square root of 5 + 20 + 15 square root of 25 + 30 square root of 5.
Simplifying this further, we get: 10 square root of 5 + 20 + 15 square root of 25 + 30 square root of 5 = 30 square root of 5 + 20 square root of 5 + 20 + 15 square root of 25.
The square root of 25 is 5, so we have: 30 square root of 5 + 20 square root of 5 + 20 + 15(5) = 30 square root of 5 + 20 square root of 5 + 20 + 75 = 50 square root of 5 + 95.
Denominator: (square root of 5 - 2)(square root of 5 + 2) = (5 - 4) = 1.
So, the expression after rationalizing the denominator becomes:
(50 square root of 5 + 95) / 1, which simplifies to 50 square root of 5 + 95.
Therefore, in the format a + b square root of 5, a is 95 and b is 50.