write each problem in form ax^p +bx^q p and q are rational numbers
[x^2-4x^(1/2)]/2x^(1/3)
the answer is 1/2x^(5/3)-2x^(1/6)
separate into two fractions
= x^2 /(2x^1/3)) - 4x^(1/2) / (2x^(1/2))
remember in dividing powers, we keep the base and subtract the exponents
= x^(5/3) / 2 - 2^(1/6)
= (1/2)x^*5/3) + 2x^(1/6)
(2 - 1/3 = 6/3-1/3 = 5/3
and
1/2 - 1/3 = 3/6 - 2/6 = 1/6 )
fourth line should say
= x^(5/3) / 2 - 2x^(1/6)
but i don't know how to get there
To write the given expression [x^2-4x^(1/2)]/2x^(1/3) in the form ax^p + bx^q, where p and q are rational numbers, we need to simplify the expression and express all the terms with rational exponents.
1. Start by simplifying each term individually.
- The numerator, x^2 - 4x^(1/2), cannot be simplified further.
- The denominator, 2x^(1/3), can be written as 2 * x^(1/3).
2. Rewrite the expression with rational exponents.
The expression becomes (x^2 - 4x^(1/2)) / (2 * x^(1/3)).
3. Combine the terms in the numerator.
The numerator contains two terms: x^2 and -4x^(1/2). Since the exponents have different values, we keep them separate.
4. Write each term with a common denominator.
The denominator is 2 * x^(1/3). We need to express each term in the numerator with the same denominator.
For x^2, we can rewrite it as (x^2 * x^(2/6)) to get a common denominator of 6.
For -4x^(1/2), we can rewrite it as (-4x^(1/2) * x^(3/6)) to get a common denominator of 6.
5. Combine the terms with a common denominator.
Now that we have a common denominator of 6, we can add the terms in the numerator.
(x^2 * x^(2/6)) - (4x^(1/2) * x^(3/6)) / (2 * x^(1/3)).
6. Simplify the terms in the numerator.
Using the properties of exponents, x^2 * x^(2/6) becomes x^(12/6 + 2/6) = x^(14/6).
-4x^(1/2) * x^(3/6) becomes -4x^(9/6) = -4x^(3/2).
7. Rewrite the expression with the simplified numerator.
The expression becomes (x^(14/6) - 4x^(3/2)) / (2 * x^(1/3)).
8. Simplify the denominator.
The denominator 2 * x^(1/3) can be written as 2 * x^(2/6), which simplifies to 2 * x^(1/6).
9. Rewrite the expression with the simplified denominator.
The expression becomes (x^(14/6) - 4x^(3/2)) / (2 * x^(1/6)).
10. Arrange the terms in ascending order.
To get the expression in the form ax^p + bx^q, where p and q are rational numbers, we can arrange the terms in ascending order of their exponents.
(x^(14/6) - 4x^(3/2)) / (2 * x^(1/6)) can be rearranged as -4x^(3/2) + x^(14/6) / (2 * x^(1/6)).
11. Simplify the expression further, if possible.
The expression -4x^(3/2) + x^(14/6) / (2 * x^(1/6)) cannot be simplified any further.
Therefore, the given expression [x^2-4x^(1/2)]/2x^(1/3) can be written in the form ax^p + bx^q as -4x^(3/2) + x^(14/6) / (2 * x^(1/6)).