Simplify
5^1/3 /2(3)^1/2 - 2^1/2
To simplify the expression, we'll start by breaking it down into smaller parts.
Let's begin with the denominator, which is 2(3)^1/2 - 2^1/2. We notice that both terms have square roots, and we can simplify them separately before subtracting.
For the first term, 2(3)^1/2, we can simplify it by multiplying 2 by the square root of 3.
2(3)^1/2 = 2 * √3 = 2√3
Now, let's simplify the second term, 2^1/2. This term can be rewritten as the square root of 2.
2^1/2 = √2
Now, we can substitute these simplified expressions back into our original expression:
5^1/3 / (2√3 - √2)
Next, let's simplify the numerator, which is 5^1/3. To do this, we need to understand what the exponent 1/3 means.
The exponent 1/3 represents taking the cube root of a number. In this case, it means taking the cube root of 5.
So, 5^1/3 is equal to the cube root of 5.
Now, we can rewrite the expression as:
∛5 / (2√3 - √2)
To simplify it further, we need to eliminate the radicals in the denominator by rationalizing the denominator.
To do this, we multiply the numerator and denominator by the conjugate of the denominator, which is the same expression but with the opposite sign in the middle.
The conjugate of (2√3 - √2) is (2√3 + √2).
By multiplying the numerator and denominator by this conjugate, we get:
(∛5 / (2√3 - √2)) * ((2√3 + √2) / (2√3 + √2))
Next, let's simplify the numerator using properties of radicals:
∛5 * (2√3 + √2) = 2∛5√3 + ∛5√2
Now, let's multiply the denominators using the distributive property:
(2√3 - √2) * (2√3 + √2) = (2√3)^2 - (√2)^2 = 4*3 - 2 = 12 - 2 = 10
Therefore, the simplified expression is:
(2∛5√3 + ∛5√2) / 10
You can also rewrite it as:
(∛5/5)(2√3 + √2)
And that is the simplified form of the expression.