evaluate 2/2root 2+3 /root 2/2root 2-3
To evaluate the given expression, let's simplify each part step by step.
1) Simplifying the numerator: 2 / (2√2)
To simplify this, multiply the numerator and denominator by the conjugate of the denominator, which is √2.
(2 / (2√2)) * (√2 / √2) = (2√2) / (2*√2*√2) = (2√2) / (2√4) = (2√2) / (2*2) = √2 / 2
2) Simplifying the denominator: √2 / (2√2 - 3)
We can rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is (2√2 + 3).
(√2 / (2√2 - 3)) * ((2√2 + 3) / (2√2 + 3)) = (√2 * (2√2 + 3)) / ((2√2 - 3) * (2√2 + 3)) = (2√4 + 3√2) / (4*2 - 3*3) = (2√4 + 3√2) / (8 - 9) = (2√4 + 3√2) / (-1) = -(2√4 + 3√2)
Now, we can substitute the simplified numerator and denominator back into the original expression:
(√2 / 2) / (-(2√4 + 3√2))
We cannot simplify this any further since the numerator and denominator do not share any common factors. Therefore, the evaluated expression is (√2 / 2) / (-(2√4 + 3√2)).
To evaluate the given expression, let's break it down step-by-step:
Step 1: Simplify the fraction 2/√2 + 3
To add fractions, we need a common denominator. The denominator of the first fraction is √2.
Multiplying the numerator and denominator of 2/√2 by √2 gives us:
2/√2 * √2/√2 = 2√2 / 2 = √2
Therefore, the expression becomes √2 + 3.
Step 2: Simplify the fraction √2 / (2√2 - 3)
To divide fractions, we need to multiply by the reciprocal of the divisor. In this case, we have:
√2 / (2√2 - 3) * (1 / (2√2 - 3))
By multiplying the numerators and denominators:
√2 / (2√2 - 3) * (1 / (2√2 - 3)) = √2 / ((2√2 - 3)(2√2 - 3))
Step 3: Simplify the denominator further.
To simplify the denominator, we can use the formula (a - b)(a + b) = a^2 - b^2.
In this case, a = 2√2 and b = 3.
(2√2 - 3)(2√2 - 3) = (2√2)^2 - 3^2
= (4 * 2) - 9
= 8 - 9
= -1
Step 4: Substitute the simplified denominator and calculate the final result.
Now we can substitute the simplified denominator (-1) back into the expression:
√2 / ((2√2 - 3)(2√2 - 3)) = √2 / -1
= -√2
So the final value of the expression is -√2.