find the value of 8ab(a^2+b^2),if a+b=r00t10 and a-b=2root2
To find the value of 8ab(a^2+b^2), we need to substitute the given values of a and b. Let's start by solving the given equations: a+b = √10 and a-b = 2√2.
To solve these equations, we can use the method of substitution:
From a+b = √10, we can solve for a:
a = √10 - b
Substituting the value of a in the second equation:
(√10 - b) - b = 2√2
√10 - 2b = 2√2
Now, isolate the value of b:
-2b = 2√2 - √10
-2b = 2√2 - √10
Divide both sides of the equation by -2:
b = (2√2 - √10)/-2
b = (√2 - √10/2)
Now, substitute the value of b back into the equation a = √10 - b:
a = √10 - (√2 - √10/2)
a = √10 - √2 + √10/2
Now that we have the values of a and b, we can calculate the value of 8ab(a^2+b^2):
8ab(a^2+b^2) = 8(√10 * (√10 - √2 + √10/2))(√10 - √2 + √10/2)^2 + (√2 - √10/2)^2)
Simplifying the expression step by step:
8ab = 8(√10 * (√10 - √2 + √10/2)) = 8(√10 * (√10/2 + √10/2 - √2))
(a^2+b^2) = ((√10 - √2 + √10/2))^2 + (√2 - √10/2))^2
= (√10 - √2 + √10/2) * (√10 - √2 + √10/2) + (√2 - √10/2) * (√2 - √10/2)
Now we can simplify:
8(√10 * (√10/2 + √10/2 - √2)) = 8(√10 * (√10 + √10 - 2√2)/2)
= 8(√10 * (2√10 - 2√2)/2)
= 4√10 * (√10 - √2)
(√10 - √2 + √10/2) * (√10 - √2 + √10/2) + (√2 - √10/2) * (√2 - √10/2)
= (√10 - √2)^2 + (√10/2)^2 + (√2)^2 + (√2 - √10/2)^2
= (10 - 2√20 + 2) + (10/4) + 2 + (2 - √20 + √10/2)
= 14 - 2√20 + 10/4 + 2 + 2 - √20 + √10/2
= 20 - 4√20 + 2.5 + 2 + 2 - √20 + √10/2
= 26.5 - 4√20 - √20 + √10/2
Now multiply the two simplified expressions:
8ab(a^2+b^2) = 4√10 * (√10 - √2) * (26.5 - 4√20 - √20 + √10/2)
At this point, you can use a calculator to compute the numerical value of 8ab(a^2+b^2).