if root a minus root b = root18 - root6 nd multiply root5 then prove a+b=18
I get this far
√a-√b = √18-√6
and have trouble figuring out what "nd multiply root5" means. How about filling in the rest using math symbols?
To prove that a + b = 18 given that √a - √b = √18 - √6, we need to follow these steps:
Step 1: Square both sides of the equation √a - √b = √18 - √6 to eliminate the square roots.
(√a - √b)^2 = (√18 - √6)^2
Step 2: Expand both sides of the equation using the formula (a - b)^2 = a^2 - 2ab + b^2.
(a - 2√ab + b) = (18 - 2√18√6 + 6)
Step 3: Simplify the equation on both sides.
a - 2√ab + b = 18 - 2√108 + 6
Step 4: Simplify the square root of 108.
a - 2√ab + b = 18 - 6√3
Step 5: Since the equation a - 2√ab + b = 18 - 6√3 is an algebraic equation, we can compare the coefficients on each side of the equation.
From the left side, the coefficient of the square root (√ab) term is -2, and from the right side, it is -6.
Therefore, -2 = -6.
Step 6: Now, setting the coefficients of √ab equal on both sides, we have:
-2 = -6
Multiplying through by -1, we have:
2 = 6
This is not a valid conclusion, as the equation is not consistent. Therefore, we cannot prove that a + b = 18 based on the initial equation √a - √b = √18 - √6.
To prove that a + b = 18, let's start by squaring both sides of the equation root(a) - root(b) = root(18) - root(6) and multiplying by root(5):
(root(a) - root(b)) * root(5) = (root(18) - root(6)) * root(5)
Simplifying the left side:
root(5a) - root(5b) = root(90) - root(30)
Next, let's square both sides of this equation:
(root(5a) - root(5b))^2 = (root(90) - root(30))^2
Expanding both sides using the formula (a - b)^2 = a^2 - 2ab + b^2:
(5a - 2√(5ab) + 5b) = (90 - 2√(90) + 30)
Now, let's simplify the equation:
5a - 2√(5ab) + 5b = 90 - 2√(90) + 30
Since we're trying to prove that a + b = 18, let's isolate that term:
5a + 5b = 120 - 2√(90) + 2√(5ab)
Rearranging the equation:
5(a + b) = 120 - 2√(90) + 2√(5ab)
Now, to continue, we need to simplify the irrational terms:
2√(90) can be simplified to 2√(9*10), which equals 2 * 3√10 = 6√10.
2√(5ab) remains the same.
Substituting these values back into the equation:
5(a + b) = 120 - 6√10 + 2√(5ab)
Next, divide both sides of the equation by 5:
a + b = (120 - 6√10 + 2√(5ab)) / 5
Now, we need to use the given information that a + b = 18 in order to prove it:
Since a + b = 18, we can substitute this value into the equation:
18 = (120 - 6√10 + 2√(5ab)) / 5
Next, multiply both sides of the equation by 5 to eliminate the fraction:
90 = 120 - 6√10 + 2√(5ab)
Now, isolate the irrational terms:
6√10 - 2√(5ab) = 120 - 90
Combine like terms on the right side:
6√10 - 2√(5ab) = 30
Rearranging the terms:
6√10 = 2√(5ab) + 30
Now, let's square both sides of the equation to eliminate the square roots:
36 * 10 = (2√(5ab) + 30)^2
Simplifying the right side:
360 = 4(5ab) + 4(2√(5ab) * 30) + 900
Further simplifying:
360 = 20ab + 240√(5ab) + 900
Now, subtract 360 from both sides:
20ab + 240√(5ab) + 900 - 360 = 0
20ab + 240√(5ab) + 540 = 0
Divide the entire equation by 20:
ab + 12√(5ab) + 27 = 0
Note that this equation should hold true regardless of the values of a and b. To prove that a + b = 18, we need to prove that the given equation satisfies this equation for any values of a and b. Unfortunately, it seems that the given equation root(a) - root(b) = root(18) - root(6) does not lead to the conclusion that a + b = 18 when following the above steps. It's possible that there was an error in the equation or the original statement. Therefore, we cannot prove that a + b = 18 based on the given information.