3(1+x)^ 1/3 -x(1-x)^ -2/3
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(1+x) sqrt 2/3
please help
the line means divisionn of the underlined numbers and variables.
To simplify the expression, let's break it down step by step:
Step 1: Simplify the numerator
The numerator of the expression is 3(1+x)^(1/3) - x(1-x)^(-2/3).
Starting with the first term, we have 3(1+x)^(1/3). To simplify this, we can use the property of fractional exponents:
(a^m)^n = a^(m*n)
So, applying this property, we can rewrite the first term as (1+x)^(1/3 * 1) = (1+x)^(1/3).
Moving on to the second term, we have -x(1-x)^(-2/3). Similarly, we can rewrite this as -x(1-x)^(-2/3 * 1) = -x(1-x)^(-2/3).
Step 2: Simplify the denominator
The denominator of the expression is (1+x) sqrt(2/3).
To simplify this, we can first simplify the square root of 2/3. The square root can be written as a fractional exponent:
sqrt(a) = a^(1/2)
So, we can rewrite sqrt(2/3) as (2/3)^(1/2).
Step 3: Put the numerator and denominator together
Now, we can rewrite the expression as:
[(1+x)^(1/3) - x(1-x)^(-2/3)] / [(1+x)(2/3)^(1/2)]
Step 4: Simplify further
Next, we need to simplify the expressions with fractional exponents.
For the numerator, we have (1+x)^(1/3). This cannot be simplified further.
For the denominator, we have (2/3)^(1/2). To simplify this, we can use the property of fractional exponents again:
(a^m)^n = a^(m*n)
So, (2/3)^(1/2) can be rewritten as 2^(1/2) / 3^(1/2). Now, the square root of 2 and 3 cannot be simplified any further.
Step 5: Combine the terms
To combine the terms, we need a common denominator for the numerator.
Multiplying the first term by (2/2) and the second term by (3/3), we get:
[(2/2)(1+x)^(1/3) - 3x(1-x)^(-2/3)] / [(1+x)(2/3)^(1/2)]
[(2(1+x)^(1/3) - 3x(1-x)^(-2/3)) / (2^(1/2)(1+x)(3^(1/2))]
Step 6: Simplify the numerator
The numerator now becomes: 2(1+x)^(1/3) - 3x(1-x)^(-2/3).
This cannot be simplified any further, so we leave it as it is.
Step 7: Finalize the expression
The final simplified expression is:
[2(1+x)^(1/3) - 3x(1-x)^(-2/3)] / [2^(1/2)(1+x)(3^(1/2))]