Question

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$6 x^{1 / 2}(2 x+3)+x^{3 / 2} \cdot 8 \quad x \geq 0$$

Short answer

The fully factored expression is \(2x^{1/2} (10x + 9)\).

Step-by-step solution

Identify Common Factors

First, identify the common factors in each term of the expression. The given expression is: \[ 6 x^{1 / 2}(2 x+3)+x^{3 / 2} \times 8 \] Notice that both terms in the expression share a common factor of \(2 \times x^{1/2}\).

Factor Out the Common Factor

Extract the common factor \(2x^{1/2}\) from each term. Doing this, we get: \[ 2x^{1/2} [3(2x+3) + 4x] \].

Simplify Inside the Bracket

Simplify the expression inside the brackets: \[ 2x^{1/2} [6x + 9 + 4x] \]. Combine like terms: \[ 2x^{1/2} [10x + 9] \].

Rewrite with Positive Exponents

Ensure that all exponents are positive. Here, all exponents are already positive, so the factored expression is simple as follows: \[ 2x^{1/2} (10x + 9) \].

Key concepts

Common Factors

To successfully factor an expression, the first step is identifying common factors. A common factor is a number or variable that divides each term in the expression without any remainder.
In the exercise example, the expression given is: \[ 6 x^{1 / 2}(2 x+3) + x^{3 / 2} \times 8 \]
You can see that both terms feature the variable \(x\) raised to a power. The lowest power, which both terms share, is \(x^{1/2}\). Additionally, 2 is common in both terms' coefficients.
So, the common factor for the entire expression is \(2 \times x^{1/2}\).
Identifying common factors simplifies solving and helps with reducing complex expressions into simpler forms.

Positive Exponents

Exponents describe how many times a number or variable is multiplied by itself. Positive exponents indicate straightforward multiplication.
For example: \[ x^{1/2} \] can be read as 'the square root of x' which is essential in both algebra and calculus.
When factoring expressions, make sure all exponents in your final answer remain positive. This keeps the expression easy to understand and manipulate.
In our exercise, the final factored expression is: \[ 2x^{1/2} (10x + 9) \]
Here, all exponents are positive, ensuring the expression is simple and ready for further calculus operations or algebraic manipulation.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This often includes:

• Combining like terms
• Factoring out common factors
• Rewriting complex fractions or roots

In our exercise, after factoring out \(2x^{1/2}\) in Step 2, we simplified the remaining expression:
\[ 2x^{1/2} [3(2x + 3) + 4x] \]
By further combining the terms inside the brackets, we get: \[ 2x^{1/2} (10x + 9) \]
This process of simplification helps us to deal with less complex forms and is crucial for solving higher-level calculus problems efficiently.

Calculus Foundations

Understanding how to factor and simplify expressions lays a strong foundation for calculus.
Calculus often requires manipulating expressions to find limits, derivatives, and integrals.
In the given exercise, we factored and simplified the expression to \[ 2x^{1/2} (10x + 9) \] which can now be easily used in various calculus techniques, like differentiation or integration.
Grasping these basic concepts ensures you can handle more complex calculus tasks, making your work much easier and more efficient. A strong foundation in algebraic manipulation will greatly benefit your learning and application of calculus principles.