Problem 34 - Precalculus: Functions and Graphs

Question

Simplify. \(\left(-6 x^{7 / 5}\right)\left(2 x^{8 / 5}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(-12x^3\).

Step 1: Multiply the Constants

Start by multiplying the constant terms in the expression. We have \((-6)\) and \(2\). Therefore, \((-6) \times 2 = -12\). The expression now becomes \(-12 \times x^{7/5} \times x^{8/5}\).

Step 2: Combine the Exponents

Apply the multiplication rule for exponents, which states that \(a^m \times a^n = a^{m+n}\).Here, we have \(x^{7/5} \times x^{8/5}\), so we add the exponents: \(7/5 + 8/5 = 15/5\). Simplify \(15/5\) to \(3\). Thus, the expression becomes \(-12x^3\).

Key Concepts

Exponent Rules

Exponent rules are essential when simplifying expressions that include powers of the same base. When you multiply terms with the same base, such as \(x^m\) and \(x^n\), you follow the product of powers rule. This rule states that you add the exponents together:

• \(x^m \times x^n = x^{m+n}\)

In the exercise, we applied this rule to combine the exponents of \(x^{7/5}\) and \(x^{8/5}\). By adding the exponents together, we obtain \(x^{(7/5 + 8/5)} = x^{15/5}\), which simplifies to \(x^3\).
This step significantly reduces the complexity of the expression. Remembering this exponent rule can make multiplying similar bases less daunting, as it only requires a simple addition of exponents.
Understanding how to manipulate exponents effectively will give you a strong foundation in algebra and help you tackle more complex mathematical problems.

Multiplication of Constants

Multiplying constants is one of the simpler steps in many algebraic problems, yet it’s crucial for simplifying expressions correctly. When you have an expression with multiple constant terms, you simply multiply them as ordinary numbers:

• Consider constants \(-6\) and \(2\) in the given problem.
• Multiply these constants: \(-6 \times 2 = -12\).

This operation reduces the expression to \(-12 \times x^{7/5} \times x^{8/5}\).
Accuracy in this step ensures that your constants carry through the simplification process correctly, maintaining the integrity of your final expression.
Multiplying the constants first in an expression keeps the process organized and helps prevent mistakes. It’s like setting a solid foundation before working on the more complex parts of the problem.

Combining Like Terms

Combining like terms helps simplify expressions by reducing the number of terms to the simplest form possible. Like terms are terms that have the same variable raised to the same power. Only their coefficients can differ.
In the context of the exercise, \(-12x^{7/5}\) and \(2x^{8/5}\) were like terms with the base \(x\), which we needed to simplify using exponent rules first before combining them into one term:

• After applying exponent rules, the powers of \(x\) were combined to \(x^3\).
• The remaining expression \(-12x^3\) is fully simplified and showcases the combined like terms.

Combining like terms is crucial for simplifying algebraic expressions and making them easier to work with in further mathematical operations. It reduces clutter and allows for a clearer understanding of the expression's behavior. Always be on the lookout for opportunities to combine terms that share the same variable and exponent.