Question

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$3 x^{-1 / 2}+\frac{3}{2} x^{1 / 2} \quad x>0$$

Short answer

The factored expression is \[3 x^{-1 / 2} (1 + \frac{1}{2} x)\].

Step-by-step solution

Identify Common Factors

Examine the terms in the given expression to identify a common factor. The expression is:\[3 x^{-1 / 2} + \frac{3}{2} x^{1 / 2}\]Both terms share the common factor of \[x^{-1/2}\] and \[3\].

Factor Out the Common Factor

Factor out the common factor \[3 x^{-1 / 2}\] from each term in the expression:\[3 x^{-1 / 2} + \frac{3}{2} x^{1 / 2} = 3 x^{-1 / 2} (1 + \frac{1}{2} x)\]

Simplify the Expression

Rewrite the factored expression to ensure all exponents are positive:\[3 x^{-1 / 2} (1 + \frac{1}{2} x)\]Since \(3 x^{-1/2}\) and \(x^{1/2}\) are already simplified as positive exponents in the coefficient \(1 + \frac{1}{2} x\), nothing further needs to be done.

Key concepts

Positive Exponents

When working with exponents, it's often important to convert them to positive values. This makes the expressions easier to understand and manipulate in calculus. If an exponent is negative, we can rewrite it using its positive counterpart.
For example, the term \( x^{-1/2} \) can be written as \( \frac{1}{x^{1/2}} \). Similarly, \( x^{-1} \) is the same as \( \frac{1}{x} \).
In expressions where these negative exponents appear, we can factor them out to simplify the expression. For instance, factoring \( 3 x^{-1/2} \) from \( 3 x^{-1/2} + \frac{3}{2} x^{1/2} \) helps us manage these terms more easily.

Common Factor

A common factor is a number or expression that is common across multiple terms in an expression. Identifying a common factor is a critical step in simplifying expressions.
When looking at the terms \( 3 x^{-1/2} \) and \( \frac{3}{2} x^{1/2} \), we can see that both terms share the factors \( 3 \) and \( x^{-1/2} \).
Factoring these out simplifies the expression and allows us to see the underlying structure more clearly:
\[ 3 x^{-1/2} + \frac{3}{2} x^{1/2} = 3 x^{-1/2} (1 + \frac{1}{2} x) \]
This is easier to work with in further calculations or when integrating and differentiating.

Calculus

Calculus involves the study of change and motion, governed by derivatives and integrals. Handling exponents accurately is pivotal in calculus, as many functions can be represented with exponential terms.
In the context of factoring expressions in calculus, simplification by factoring makes differentiation or integration straightforward. For example, the simplified form \( 3 x^{-1/2} (1 + \frac{1}{2} x) \) is easier to differentiate or integrate compared to the original expression.
Factoring out common factors isn't just a simplification tool but a step to make the expressions more manageable for further mathematical operations employed in calculus.