Question

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$4(3 x+5)^{1 / 3}(2 x+3)^{3 / 2}+3(3 x+5)^{4 / 3}(2 x+3)^{1 / 2} \quad x \geq-\frac{3}{2}$$

Short answer

\[(3 x+5)^{1/3}(2 x+3)^{1/2}(17x + 27)\]

Step-by-step solution

Identify the Common Factor

First, examine both terms in the expression to identify any common factors. In the expression \[4(3 x+5)^{1 / 3}(2 x+3)^{3 / 2}+3(3 x+5)^{4 / 3}(2 x+3)^{1 / 2} \] both \((3 x+5)^{1 / 3}\) and \((2 x+3)^{1 / 2}\) are common factors.

Factor Out the Common Terms

Extract the smallest power of the common factors from each term. This yields: \((3x + 5)^{1/3}\) and \((2x + 3)^{1/2}\).

Rewrite Each Term with Common Factors

With the identified common factors factored out, the expression becomes: \[(3 x+5)^{1 / 3}(2 x+3)^{1 / 2} [4(2 x+3) + 3(3 x+5)]\]

Simplify the Remaining Expression

Simplify the expression inside the brackets: \[4(2 x+3) + 3(3 x+5) = 8x + 12 + 9x + 15 = 17x + 27\]

Rewrite the Final Factored Expression

Reassemble the expression using the common factors, yielding the final factorization: \[(3 x+5)^{1 / 3}(2 x+3)^{1 / 2} (17x + 27)\]

Key concepts

Common Factors

When factoring expressions, the first step is finding the common factors. A common factor is a term that appears in each part of the expression. This can be a number, a variable, or a combination of both that can be factored out.

For example, let's look at the expression:

• \begin{align*}4(3 x+5)^{1 / 3}(2 x+3)^{3 / 2}+3(3 x+5)^{4 / 3}(2 x+3)^{1 / 2}ewlineewlineewlineewlineewlineewlineewlineewlineewlineewline Here, we see that \((3x + 5)^{1/3}\) and \((2x + 3)^{1/2}\) are present in both terms.
  
  To factor out, use the smallest exponent of the common factor, which gives: \((3x+5)^{1/3}\) and \((2x+3)^{1/2}\). Factoring correctly simplifies working with the expression later.

Radical Expressions

Radical expressions involve roots, which can sometimes make equations look difficult and complex. However, breaking them down into parts we know makes them easy to handle.

In our example, ewlineewlineewline \((3 x+5)^{1 / 3}\) is a cube root operation. Similarly, \((2 x+3)^{1 / 2}\) represents a square root.

While solving, ensure to simplify radical expressions by factoring out roots first. This process helps rewrite terms in a simpler and more manageable form.

Therefore, we rewrite our expression with the common factors factored out, which makes the remaining expression easier to handle.

Calculus Preparation

Factoring radical expressions with positive exponents is a skill you'll frequently use in calculus.

Learning how to manage and simplify such expressions helps in solving complex integrals, derivatives, and various calculus problems efficiently.

In our example, breaking down the given terms ewlineewline and appropriately factoring out \((3 x+5)^{1 / 3}(2 x+3)^{1 / 2}\) makes it easier to work with polynomial expressions.

This systematic approach and attention to detail foster the kind of problem-solving mindset you need in calculus.

Positive Exponents

After factoring, it's crucial to express all terms with positive exponents. Positive exponents are easier to interpret and solve.

In our example, we start with terms that have positive rational exponents already:ewlineewline \((4(3 x+5)^{1 / 3}(2 x+3)^{3 / 2}+3(3 x+5)^{4 / 3}(2 x+3)^{1 / 2})\)
Finally, our factored form \((3 x+5)^{1/3}(2 x+3)^{1 / 2}(17x + 27)\) keeps all the exponents positive.

Keeping exponents positive simplifies calculations and ensures clarity, facilitating further steps in any algebraic or calculus problem.