Question

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

Short answer

\frac{1}{x^2 (x^2-1)^{1/2}}

Step-by-step solution

Simplify the Numerator

The given expression is: \[ \frac{\frac{x^2}{(x^2-1)^{1/2}} - (x^2-1)^{1/2}}{x^2}\]First, simplify the numerator which is \[ \frac{x^2}{(x^2-1)^{1/2}} - (x^2-1)^{1/2}\].

Combine Like Terms in the Numerator

Combine the two terms in the numerator over a common denominator:\[ \frac{x^2}{(x^2-1)^{1/2}} - (x^2-1)^{1/2} = \frac{x^2 - (x^2-1)}{(x^2-1)^{1/2}}\]Simplify the expression inside the numerator.

Simplify the Expression in the Numerator

Simplify the numerator: \[ x^2 - (x^2 - 1) = x^2 - x^2 + 1 = 1\].So the expression becomes: \[ \frac{1}{(x^2-1)^{1/2}}\].

Incorporate the Denominator

Now, divide this result by the denominator of the original fraction \[x^2\]:\[ \frac{\frac{1}{(x^2-1)^{1/2}}}{x^2} = \frac{1}{x^2 \cdot (x^2-1)^{1/2}}\].

Simplify the Denominator

Combine the denominator terms: \[ \frac{1}{x^2 \cdot (x^2-1)^{1/2}} = \frac{1}{x^2 (x^2-1)^{1/2}}\].

Key concepts

Simplifying Expressions

Simplifying expressions often involves breaking down a complex expression into simpler parts. Here, the goal is to express the given algebraic fraction in its simplest form.
In our exercise, we start with the expression \[ \frac{\frac{x^2}{(x^2-1)^{1/2}} - (x^2-1)^{1/2}}{x^2} \].
First, we tackle the numerator: \[ \frac{x^2}{(x^2-1)^{1/2}} - (x^2-1)^{1/2} \].
By finding a common denominator, we can combine these terms into a single fraction: \[ \frac{x^2 - (x^2-1)}{(x^2-1)^{1/2}} \].
This step simplifies the expression inside the numerator. Next, we simplify \[ x^2 - (x^2 - 1) = 1 \], leading to the simpler form: \[ \frac{1}{(x^2-1)^{1/2}} \].

Positive Exponents

In algebra, positive exponents are much easier to work with than negative ones. Positive exponents indicate multiplication, while negative exponents often involve division or reciprocals.
In simplifying our fraction, we aim to ensure all exponents are positive.
Initially, the expression has an exponent of \[ \left(x^2-1\right)^{1/2} \] which is already positive.
As we simplify further, we keep this principle intact to arrive at: \[ \frac{1}{x^2 (x^2-1)^{1/2}} \].
Notice how keeping positive exponents helps in understanding and simplifying the expression.

Radicals

Radicals or roots often appear in algebra and calculus. We find square roots (like \[ \sqrt{a} \]), cube roots, and so on.
In our exercise, \[ \left(x^2-1\right)^{1/2} \] represents a square root. Simplifying expressions with radicals often involves rationalizing them to avoid roots in the denominator.
Here though, we end up with a radical in the denominator, but in simplified form: \[ \frac{1}{x^2 (x^2-1)^{1/2}} \].
Understanding how to manipulate and simplify radicals is key in algebraic expressions and problem-solving.