Solve
(root10+root18)/(root8+(root(3-root5)))
Urgent
(√10+√18)/(2√2+√(3-√5))
Note that
3-√5 = (5-2√5+1)/2 = 1/2 (√5-1)^2
so now you have
(√10+√18)/(2√2 + 1/√2 (√5-1))
√2(√5+3) /(√2 (2 + (√5-1)/2))
2(√5+3)/(4+√5-1)
2(3+√5)/(3+√5)
2
(sqrt 5 sqrt 2 + 3 sqrt 2) / [ 2 sqrt 2 + sqrt ( 3 - sqrt 5) ]
sqrt 2(sqrt 5 + 3 ) / [ 2 sqrt 2 + sqrt ( 3 - sqrt 5) ]
I am suspicious that you mean
(sqrt 5 sqrt 2 + 3 sqrt 2) / [ 2 sqrt 2 * sqrt ( 3 - sqrt 5) ]
then
sqrt 2(sqrt 5 + 3 ) / [ 2 sqrt 2* sqrt ( 3 - sqrt 5) ]
(1/2) (3+sqrt 5)/ sqrt(3-sqrt 5)
multiply top and bottom by sqrt (3+sqrt5)
(1/2) (3+sqrt 5)^(3/2) / sqrt(3-sqrt 5)sqrt(3+sqrt 5)
(1/2) (3+sqrt 5)^(3/2) / sqrt(9-5)
(1/4) (3+sqrt 5)^3/2
To begin solving the expression (root10 + root18) / (root8 + (root(3-root5))), let's simplify it step by step:
Step 1: Simplify the numerator:
root10 can be written as root5 * root2, and root18 can be written as root9 * root2. So, the numerator can be rewritten as:
(root5 * root2 + root9 * root2)
We can factor out root2:
(root2 * (root5 + root9))
Step 2: Simplify the denominator:
root8 can be written as root4 * root2, and root(3-root5) cannot be simplified further. So, the denominator can be written as:
(root4 * root2 + root(3-root5))
Step 3: Substitute the simplified forms into the original expression:
(root2 * (root5 + root9)) / (root4 * root2 + root(3-root5))
Step 4: Simplify further by canceling out common terms:
The root2 terms in the numerator and denominator cancel out, so we are left with:
(root5 + root9) / (root4 + root(3-root5))
And that's the simplified form of the expression.
To solve the expression (root10+root18)/(root8+(root(3-root5))), we can simplify it step by step:
Step 1: Simplify the numerator.
- The square root of 10 can be written as √10.
- The square root of 18 can be written as √18.
So, the numerator becomes √10 + √18.
Step 2: Simplify the denominator.
- The square root of 8 can be written as √8.
- The square root of (3 - √5) can be left as it is.
So, the denominator becomes √8 + √(3 - √5).
Step 3: Rationalize the denominator.
- Multiplying the expression (√8 - √(3 - √5)) to the denominator and the numerator will help to rationalize the denominator.
The expression becomes:
((√10 + √18) * (√8 - √(3 - √5))) / (√8 + √(3 - √5)) * (√8 - √(3 - √5))
Step 4: Simplify the numerator.
- Multiply (√10 + √18) with (√8 - √(3 - √5)).
The numerator becomes (√10 * √8) + (√10 * (-√(3 - √5))) + (√18 * √8) + (√18 * (-√(3 - √5))).
Step 5: Simplify the denominator.
- Multiply (√8 + √(3 - √5)) with (√8 - √(3 - √5)).
The denominator becomes (√8 * √8) + (√8 * (-√(3 - √5))) + (√(3 - √5) * √8) + (√(3 - √5) * (-√(3 - √5))).
Step 6: Simplify the expression.
- Simplify the multiplied terms in the numerator and denominator.
The expression becomes:
(√80 - √10√(3 - √5) + √144 - √18√(3 - √5)) / (8 + (-√(3 - √5)) + (-√(3 - √5)) + (3 - √5)).
Step 7: Combine like terms.
- Combine the like terms in the numerator and denominator.
The expression becomes:
(√80 + √144 - √10√(3 - √5) - √18√(3 - √5)) / (8 - 2√(3 - √5) + 3 - √5).
Step 8: Simplify the numerator further.
- Simplify (√80 + √144) since they both are perfect squares.
The numerator becomes:
(8√5 + 12) - √10√(3 - √5) - √18√(3 - √5).
Step 9: Simplify the denominator further.
- Simplify (8 - 2√(3 - √5) + 3 - √5).
The denominator becomes:
11 - 2√(3 - √5) - √5.
Step 10: Combine the final expression.
- Combine the simplified numerator and denominator.
The final expression becomes:
(8√5 + 12 - √10√(3 - √5) - √18√(3 - √5)) / (11 - 2√(3 - √5) - √5).
This is the simplified form of the given expression.