Question

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

Short answer

\( \frac{3x + 2}{(1+x)^{1/2}} \)

Step-by-step solution

Identify Common Terms

Recognize that both terms in the given expression \(\frac{x}{(1+x)^{1 / 2}}+2(1+x)^{1 / 2}\) include \( (1+x)^{1/2} \). This will help in combining them into a single quotient.

Rewriting the Expression

Rewrite each term to have the common term \( (1+x)^{1/2} \) in the denominator. Notice that \( 2(1+x)^{1/2} \) can be written as \( \frac{2(1+x)}{(1+x)^{1/2}} \).

Simplify the Exponents

Simplify the exponents in the numerator by combining the terms: \( \frac{x}{(1+x)^{1/2}} + \frac{2(1+x)^{1}}{(1+x)^{1/2}} \) becomes \( \frac{x}{(1+x)^{1/2}} + \frac{2(1+x)^{1/2 + 1/2}}{(1+x)^{1/2}} \).

Combine the Fractions

Combine the fractions under a single common denominator to get: \( \frac{x + 2(1+x)}{(1+x)^{1/2}} \).

Simplify the Numerator

Expand and combine the terms in the numerator: \( x + 2(1+x) = x + 2 + 2x = 3x + 2 \).

Final Expression

Write the simplified expression as a single quotient: \( \frac{3x + 2}{(1+x)^{1 / 2}} \).

Key concepts

positive exponents

When working with exponents, it is important to only have positive values. This is because negative exponents can make expressions harder to work with.
For example, the expression \(\frac{x}{(1+x)^{-\frac{1}{2}}}\) has a negative exponent for \( (1+x) \). We can rewrite this as a positive exponent by using the property:
\[ x^{-a} = \frac{1}{x^a} \]
By moving \( (1+x)^{-\frac{1}{2}} \) to the denominator, it becomes positive:
\[ \frac{x}{(1+x)^{-\frac{1}{2}}} = x(1+x)^{\frac{1}{2}} \]
Similarly, we always aim to rewrite expressions so that all exponents are positive.

single quotient

Combining multiple terms into a single quotient simplifies many algebraic manipulations. In our example, we start with two terms: \( \frac{x}{(1+x)^{\frac{1}{2}}} + 2(1+x)^{\frac{1}{2}} \). To convert these into a single quotient, follow these steps:

• Identify common terms. Here, both terms have \( (1+x)^{\frac{1}{2}} \).
• Rewrite the second term to make sure it shares a common base with the first term: \[ 2(1+x)^{\frac{1}{2}} = \frac{2(1+x)}{(1+x)^{\frac{1}{2}}} \]
• Now both fractions share \( (1+x)^{\frac{1}{2}} \) as a common base.

This allows us to combine them easily into one expression.

common denominator

Combining fractions requires a common denominator. For our given expression, both terms already share \( (1+x)^{\frac{1}{2}} \) as their denominator. This makes it easy to combine them: \[ \frac{x}{(1+x)^{\frac{1}{2}}} + \frac{2(1+x)}{(1+x)^{\frac{1}{2}}} = \frac{x + 2(1+x)}{(1+x)^{\frac{1}{2}}} \]

• When the denominators are the same, you can simply add the numerators together.
• This step reduces two fractions into one, which simplifies further calculation.

Finding a common denominator is a key step in combining fractions effectively.

combine fractions

Finally, we combine the fractions and simplify.
First, look at the combined numerator: \( \frac{x + 2(1+x)}{(1+x)^{\frac{1}{2}}} = \frac{x + 2 + 2x}{(1+x)^{\frac{1}{2}}} \).
This simplifies further:
\[ x + 2 + 2x = 3x + 2 \]
Now, our expression becomes: \[ \frac{3x + 2}{(1+x)^{\frac{1}{2}}} \]
This is the fully simplified single quotient with only positive exponents.
Combining fractions makes the expression easier to manage and interpret. It also lays the foundation for further simplification!