Problem 12 - Precalculus: Functions and Graphs

Question

Simplify. $$\left(-3 x^{-2}\right)\left(4 x^{4}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(-12x^2\).

Step 1: Apply the Product of Powers Property

According to the product of powers property, when you multiply two expressions with the same base, you add the exponents. Here, we have two expressions: \( -3x^{-2} \times 4x^4 \).First, factor the numerical coefficients and the powers of \(x\): Numerical: \(-3 \times 4 = -12\).Powers of \(x\): \( x^{-2} \times x^4 = x^{-2+4} = x^2 \).

Step 2: Combine the Results

Combine the results from Step 1 to arrive at the simplified expression: \(-12x^2\).

Key Concepts

Product of Powers Property

When working with algebraic expressions, especially those with exponents, it's essential to understand how to apply the **Product of Powers Property**. This rule helps in simplifying expressions where you have the same base raised to different exponents.

Here's how it works:

• Whenever you multiply terms with like bases, you add the exponents of those bases.
• For example, if you have two expressions, say, with the base of 'x', such as \(x^a\) and \(x^b\), the result would be \(x^{a+b}\).

This property simplifies the process of dealing with powers and helps to consolidate expressions into a single term.

In our problem, applying this property correctly allows us to transform \(x^{-2} \times x^4\) into \(x^{-2+4} = x^2\). This illustrates the ease of working with powers by transitioning from multiple terms to a simpler singular expression.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra, using properties like the product of powers to make complex expressions manageable. It involves bringing an expression into the simplest form possible without altering its value. Here are some general steps:

• First, tackle any operations within the expression, such as the multiplication of coefficients.
• Second, deal with the variables using exponent rules, like combining powers of the same base.
• Finally, recombine the terms into a new, simplified expression.

With our specific example, we begin by multiplying the coefficients, \(-3 \times 4\), to get \(-12\), and then apply our exponent rules to combine \(x^{-2}\) and \(x^4\) into \(x^2\). Together, these steps result in the streamlined expression \(-12x^2\).

These simplification skills are crucial, reducing potential errors and making calculations more efficient.

Negative Exponents

Understanding negative exponents is crucial for handling algebraic expressions effectively. A negative exponent indicates that the base is on the "wrong" side of a fraction, and so it should be inverted, or moved to the other side of the fraction.

To convert a term with a negative exponent:

• Flip the base and change the sign of the exponent to positive.
• For instance, \(x^{-n}\) becomes \(\frac{1}{x^n}\).

In the given expression, \(x^{-2}\) means the base \(x\) is squared and in the denominator when represented fractionally: \(\frac{1}{x^2}\). Despite dealing with such exponents in algebra involving multiplication, the rules of exponents let us handle them straightforwardly, maintaining the integrity of expressions and simplifying them accurately.
For our exercise, we eventually translate our simplification to result in a positive exponent, which is typically more practical and easier to work with in further mathematical operations.