Given that (√3-5√2)(√3+2)=a+b√6. Find a and b. Show workings bit by bit using surd method
To solve this problem, we will simplify the expression (√3-5√2)(√3+2) step by step using the surd method.
First, we will apply the difference of squares formula:
(√3-5√2)(√3+2) = (√3)^2 - (5√2)^2
Simplifying further:
= 3 - 25(2)
= 3 - 50
= -47
Therefore, a = -47 and b = 0 since there is no √6 term in the simplified expression.
To find the values of a and b in the expression (√3 - 5√2)(√3 + 2) = a + b√6, we can use the surd method.
First, let's expand the expression (√3 - 5√2)(√3 + 2):
= (√3 * √3) + (√3 * 2) + (-5√2 * √3) + (-5√2 * 2)
= 3 + 2√3 - 5√6 - 10√2
Next, we combine like terms:
= (3 - 10√2) + (2√3 - 5√6)
Now, we need to express the terms involving √2 and √6 only in terms of √6. To do this, we'll multiply the coefficients of 10√2 by √2/√2 (which equals 1) and the coefficients of 5√6 by √3/√3 (which equals 1):
= (3 - 10√2 * √2/√2) + (2√3 - 5√6 * √3/√3)
= (3 - 10√4) + (2√3 - 5√18)
Simplifying further:
= 3 - 10(2) + 2√3 - 5(3√2)
= 3 - 20 + 2√3 - 15√2
= -17 + 2√3 - 15√2
So, we have a = -17 and b = 2. Therefore, the expanded expression (√3 - 5√2)(√3 + 2) = a + b√6 can be written as -17 + 2√6.
To find the values of a and b in the expression (√3-5√2)(√3+2) = a + b√6, we will use the surd method. Here's what we need to do step by step:
Step 1: Expand the expression (√3-5√2)(√3+2).
Using the difference of squares formula, we can expand this expression as follows:
(√3-5√2)(√3+2) = (√3 * √3) + (√3 * 2) + (-5√2 * √3) + (-5√2 * 2)
Simplifying this expression, we get:
= (3) + (2√3) + (-5√2√3) + (-10√2)
Step 2: Simplify the radicals.
To simplify the expression further, we need to simplify the square roots. Remember that √a * √b = √(a * b).
We can simplify the expression -5√2√3 as follows:
-5√2√3 = -5 * √(2*3) = -5√6
Applying the same logic, we can simplify -10√2 as:
-10√2 = -10 * √2 = -10√2
After simplification, the expression becomes:
= 3 + 2√3 - 5√6 - 10√2
Step 3: Reorder the expression.
To achieve the standard form a + b√6, we reorder the terms by grouping similar radicals together and the constants separately.
= (3 - 10√2) + (2√3 - 5√6)
Step 4: Identify the values of a and b.
Comparing this with the standard form a + b√6, we find:
a = 3 - 10√2
b = 2√3 - 5√6
Therefore, the values of a and b are:
a = 3 - 10√2
b = 2√3 - 5√6