I am a bit confused about conjugates in algebra. I am supposed to multiply 8/(the square root of two + 4) by the square root of 2 -4. My math book then goes on to simplify this to (8 times the square root of two minus 32) divided by (2-4 times the square root of two plus 4 times the square root of 2 minus 16). I do not see how the bottom part came to be. Could someone offer an explanation, please?
if a=4
b=20
c=50 us tge answer 5,340.00
Certainly! I can help explain the process of simplifying the expression for you.
To simplify the expression, we need to rationalize the denominator by getting rid of the square root term. The method we can use here is multiplying both the numerator and denominator by the conjugate of the denominator.
The conjugate of a binomial expression is obtained by changing the sign between the two terms. In this case, the conjugate of the denominator (√2 + 4) is (√2 - 4). So, we multiply both the numerator and denominator of the expression by the conjugate (√2 - 4) to rationalize the denominator.
Let's go step by step:
1. Start with the expression: 8 / (√2 + 4).
2. Multiply the numerator and denominator by the conjugate of the denominator: (√2 - 4).
(8 / (√2 + 4)) * (√2 - 4)
3. To multiply the fractions, we need to multiply the numerators and denominators separately:
Numerator:
8 * (√2 - 4) = 8√2 - 32
Denominator:
(√2 + 4) * (√2 - 4) = (√2)^2 - (4)^2 = 2 - 16 = -14
4. Simplifying the expression, we have:
(8√2 - 32) / -14
As a result, the simplified expression is (8√2 - 32) / -14.
Please note that the expression in your math book seems to have a sign error. It should be divided by -14, not multiplied by -14 in the denominator.
I hope this explanation clears up any confusion you had about the simplification process! Let me know if you have any further questions.