Find a and b
(root2.3 - root0.69)/(root2.3 + root0.69)
= a + b root30
multiply by (√2.3 - √.69) top and bottom and you get
(√2.3 - √.69)^2 / ((√2.3 + √.69)(√2.3 - √.69))
= 2.3 - 2√1.587 + .69) / (2.3-.69)
= (2.99 - 2√1.587)/1.61
Now how you expect to get √30 out of that is beyond me.
Maybe you can figure it out
perhaps we have to match rationals with rationals and irrationals with irrationals?
let's give it a try, picking up at oobleck's last step
(2.99 - 2√1.587)/1.61 = a + b√30
so a = 2.99/1.61 = 13/7
b√30 = -2√.1.587 / 1.61
b = -2√.1.587 / (1.61√30) = appr -.2857
a = 13/7
b = -.2857
so a + b√30 = 13/7 - .2857√30
check:
13/7 - .2857√30 = appr .29222..
and
(2.99 - 2√1.587)/1.61 = 2.9222..
ok then!
To simplify the given expression, we will rationalize the denominator.
Step 1: Rationalize the denominator:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of sqrt(2.3) + sqrt(0.69) is sqrt(2.3) - sqrt(0.69).
Numerator: (sqrt(2.3) - sqrt(0.69)) * (sqrt(2.3) - sqrt(0.69))
Denominator: (sqrt(2.3) + sqrt(0.69)) * (sqrt(2.3) - sqrt(0.69))
Step 2: Simplify the numerator:
Using the formula (a - b)(a - b) = a^2 - 2ab + b^2, we can simplify the numerator as follows:
Numerator = (sqrt(2.3))^2 - 2(sqrt(2.3))(sqrt(0.69)) + (sqrt(0.69))^2
Numerator = 2.3 - 2(sqrt(2.3))(sqrt(0.69)) + 0.69
Step 3: Simplify the denominator:
Using the formula (a - b)(a + b) = a^2 - b^2, we can simplify the denominator as follows:
Denominator = (sqrt(2.3))^2 - (sqrt(0.69))^2
Denominator = 2.3 - 0.69
Step 4: Simplify the expression:
Now, we can simplify the given expression by dividing the numerator by the denominator:
Expression = (2.3 - 2(sqrt(2.3))(sqrt(0.69)) + 0.69) / (2.3 - 0.69)
Step 5: Represent in the form a + b(sqrt(30)):
To represent the simplified expression in the form a + b(sqrt(30)), we need to isolate sqrt(30) if it exists.
Considering that there is no sqrt(30) term in the expression, we can assume that a = 1 and b = 0.
Therefore, (sqrt(2.3) - sqrt(0.69)) / (sqrt(2.3) + sqrt(0.69)) can be represented as a + b(sqrt(30)), where a = 1 and b = 0.
To find the values of a and b in the equation
(root2.3 - root0.69)/(root2.3 + root0.69) = a + b root30,
we can use a technique called rationalizing the denominator. Rationalizing the denominator involves manipulating the expression to eliminate any square roots in the denominator.
Here's how we can do that step-by-step:
1. Multiply the numerator and the denominator by the conjugate of the denominator, which is the expression obtained by changing the sign of the second term. In this case, the conjugate of root2.3 + root0.69 is root2.3 - root0.69.
(root2.3 - root0.69)/(root2.3 + root0.69) * (root2.3 - root0.69)/(root2.3 - root0.69)
2. Simplify the numerator by multiplying using the distributive property.
= (root2.3 * root2.3) - (root2.3 * root0.69) - (root0.69 * root2.3) + (root0.69 * root0.69)
3. Simplify the denominator using the distributive property.
= (root2.3 * root2.3) - (root2.3 * root0.69) + (root0.69 * root2.3) - (root0.69 * root0.69)
4. Simplify by combining like terms. Note that root2.3 * root0.69 and root0.69 * root2.3 yield the same result.
= root2.3 * root2.3 - root0.69 * root0.69
5. Since root2.3 * root2.3 is equal to 2.3 and root0.69 * root0.69 is equal to 0.69, we can simplify further.
= 2.3 - 0.69
6. Calculate the resulting value.
= 1.61
Now, we have the following equation:
1.61 = a + b root30
To find the values of a and b, we need to make another equation by comparing the coefficients of terms without any square roots.
Comparing the coefficients, we see that:
1.61 = a (since there is no coefficient for root30 on the right-hand side)
Therefore, a = 1.61.
This means that a = 1.61 and b = 0, as there is no coefficient of root30 on the right-hand side.
So, the values of a and b are a = 1.61 and b = 0.