Question

Expressions that occur in calculus are given. Write each expression as a single quotient in which only positive exponents and radicals appear. $$\frac{\frac{1+x^{2}}{2 \sqrt{x}}-2 x \sqrt{x}}{\left(1+x^{2}\right)^{2}} \quad x>0$$

Short answer

\(\frac{1 - 3x^2}{2 \sqrt{x} (1 + x^2)^2}\).

Step-by-step solution

Simplify the numerator

Rewrite the numerator by breaking it into two terms: \(\frac{1 + x^2}{2 \sqrt{x}} - 2x \sqrt{x}\). The term \(2x \sqrt{x}\) can be rewritten using exponent rules as \(2x^{3/2}\).

Combine terms over a common denominator

The first term in the numerator is already a fraction, but the second term can be written as a fraction as well: \(\frac{2x^{3/2}}{1}\). To combine these terms, find a common denominator, which is \(2\sqrt{x}\). This gives: \[\frac{1 + x^2 - 4x^{3/2} \sqrt{x}}{2 \sqrt{x}} = \frac{1 + x^2 - 4x^2}{2 \sqrt{x}}.\]

Simplify the combined numerator

Combine like terms in the numerator: \(1 + x^2 - 4x^2 = 1 - 3x^2\). So, the numerator simplifies to \(\frac{1 - 3x^2}{2 \sqrt{x}}\).

Write the entire expression as a single quotient

Now, place the combined numerator over the original denominator: \[\frac{\frac{1 - 3x^2}{2 \sqrt{x}}}{(1 + x^2)^2}.\] Invert the denominator and multiply: \(\frac{1 - 3x^2}{2\sqrt{x} (1 + x^2)^2}\).

Simplify the expression

We now have the expression written as a single quotient with positive exponents and radicals: \[\frac{1 - 3x^2}{2 \sqrt{x} (1 + x^2)^2}.\]

Key concepts

Positive Exponents

When working with expressions in calculus, it's often useful to express terms with positive exponents.
This makes it easier to integrate or differentiate the expression later on.
An exponent indicates how many times a number (the base) is multiplied by itself.

• For example, in the term `x^2`, `x` is multiplied by itself, giving `x * x`.
• A negative exponent means you take the reciprocal of the base. For instance, `x^{-2}` translates to `1/x^2`.

In the given exercise, simplifying terms like `2x^{3/2}` helps keep exponents positive. This avoids complicating the expression unnecessarily.

Common Denominator

Combining fractions often requires a common denominator.
This involves adjusting the fractions so they share the same bottom part (denominator).

• For example, given `1/2` and `3/4`, the common denominator is 4.
• You would transform `1/2` to `2/4` so both fractions can be easily added: `2/4 + 3/4 = 5/4`.

In our exercise, the term `2x^{3/2}` needed to be rewritten as a fraction with the same denominator as `1 + x^2`.
By setting a common denominator of `2√x`, we could combine them smoothly.

Fraction Simplification

Simplifying fractions means reducing them to their simplest form.
This often involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

• For example, in the fraction `4/8`, both 4 and 8 can be divided by their GCD, which is 4. Thus `4/8` simplifies to `1/2`.

Our exercise involves breaking the numerator `1 + x^2 - 4x^2` down and simplifying common terms. By handling like terms, we achieve `1 - 3x^2`, making the whole fraction neater.

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, etc.

• The square root of a number `x` is denoted by `√x` and asks the question: `What number, multiplied by itself, gives `x`?
• For example, `√4 = 2`, because `2 * 2 = 4`.

Including radicals can make an expression a bit complex, but necessary in some cases for the sake of simplicity and understanding.
In our example, we convert `2√x` to `x^{1/2}` using exponent rules. This keeps everything formatted in a consistent, manageable way.