Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression, writing your answer using positive exponents only. $$ \left(2 x^{3}\right)\left(\frac{1}{4} x^{2}\right) $$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(\frac{1}{2}x^{5}\).

Step 1: Multiply coefficients

First, we need to multiply the coefficients of the two expressions together. In this case, we have \(2\) and \(\frac{1}{4}\). So, we will multiply these two numbers together: \[ 2\cdot\frac{1}{4}=\frac{1}{2} \] 2. Apply exponent properties:

Step 2: Apply exponent properties

Next, we will address the exponential part of the expression. We already have the terms \(x^{3}\) and \(x^{2}\) - to multiply these two expressions together, we need to remember the exponent rule: \[ a^{m} \cdot a^{n} = a^{m+n} \] In our case, \(a=x\), \(m=3\), and \(n=2\). So, our expression will look like this: \[ x^{3} \cdot x^{2} = x^{3+2} \] Now, we will add the exponents: \[ x^{3} \cdot x^{2} = x^{5} \] 3. Combine the simplified coefficients and exponent:

Step 3: Combine simplifications

Finally, let's combine the simplified coefficient and exponent from Steps 1 and 2 to obtain the simplified expression with a positive exponent: \[ \frac{1}{2} \cdot x^{5} \] Now we have the simplified expression with a positive exponent: \[ \frac{1}{2}x^{5} \]

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical formulas into a simpler, more understandable form. This makes it easier to work with and solve problems. When simplifying, you focus on combining like terms and using arithmetic operations effectively. In our exercise, we have:

• The expression \((2x^3)(\frac{1}{4}x^2)\) to simplify.
• We start by simplifying each part separately—first the coefficients, then the variables and their exponents.

By simplifying expressions, we can turn a potentially overwhelming problem into a manageable solution, ensuring our calculations remain accurate and understandable.

Positive Exponents

Using positive exponents is a standard practice in mathematics because it results in expressions that are easier to interpret and work with than negative exponents. An exponent stands for how many times a number, or a base, is multiplied by itself. In our given problem:

• We have \(x^3\) and \(x^2\).
• These exponents represent \(x\) being multiplied by itself \(3\) times and \(2\) times, respectively.

Instead of using negatives, which often indicate division, we focus only on positive exponents here to keep calculations straightforward. Combining them, we use the exponent rule:

\(x^3 \cdot x^2 = x^{3+2} = x^5\)

This approach ensures clarity and avoids errors associated with interpreting negative powers.

Multiplying Coefficients

Multiplying coefficients is one of the essential steps when simplifying expressions involving multiplication. Coefficients are the numerical parts of terms that do not contain variables. In our task:

• The coefficients are \(2\) and \(\frac{1}{4}\).
• We multiply them together, following the basic rules of multiplication:

\(2 \cdot \frac{1}{4} = \frac{2}{4} = \frac{1}{2}\)

By thinking of this in terms of simple fractions, we reduce the potential for mistakes and focus the effort on handling more complex algebraic parts. Once multiplied, combining with the variable part of the expression completes the simplification process.