Problem 44 - Algebra for College Students

Question

Simplify each expression. Express final results without using zero or negative integers as exponents. \(x^{-3} \cdot x^{-4}\)

Step-by-Step Solution

Answer

\(\frac{1}{x^7}\)

Step 1: Apply the Product of Powers Property

When multiplying expressions with the same base, you add the exponents. For the given expression, we have:\[x^{-3} \cdot x^{-4} = x^{-3 + (-4)} = x^{-7}\]

Step 2: Simplify the Expression to Remove Negative Exponents

The expression \(x^{-7}\) has a negative exponent. To express it with positive exponents, move the term with the negative exponent to the denominator:\[x^{-7} = \frac{1}{x^7}\]

Key Concepts

Product of Powers

Imagine you have a repeated multiplication involving the same base. When multiplying two powers with the same base, you can simplify the process by adding their exponents. This is known as the Product of Powers Property. If you encounter something like \(x^a \cdot x^b\), where \(x\) is the base, the rule to remember is that you simply add the exponents:

• \(x^a \cdot x^b = x^{a+b}\)

In our exercise, we had two powers with bases of \(x\): \(x^{-3} \cdot x^{-4}\). Using the Product of Powers Property, we add the exponents \(-3 + (-4)\), which simplifies to \(-7\). This gives us \(x^{-7}\). By understanding this property, multiplying powers becomes a breeze, and it's easier to manage expressions with multiple terms.

Simplification of Expressions

Simplifying expressions may sound complicated, but it often involves just a few simple steps. Once you've applied the Product of Powers Property, you might end up with an expression that has negative exponents. Simplification is all about transforming these expressions into simpler forms without changing their value. If you get an expression like \(x^{-7}\), we can manipulate it to become easier to understand or more acceptable for specific situations, like when positive exponents are needed.

• Negative exponents can be eliminated by transforming them into fractions.
• This aids in expressing the term more neatly, resulting in \(\frac{1}{x^7}\) in this case.

The process might involve moving terms, like bringing a power with a negative exponent into the denominator to make a positive exponent. This helps in achieving an equivalent, simplified expression.

Positive Exponents

Working with positive exponents is often cleaner and more intuitive, especially when dealing with simplified mathematical expressions. Positive exponents mean that you're simply multiplying the base by itself a certain number of times, as specified by the exponent. For the expression \(x^{-7}\), having a negative exponent initially might look a little confusing, but it simply represents the reciprocal of the base raised to a positive exponent. By rewriting \(x^{-7}\) as \(\frac{1}{x^7}\), we effectively express the same mathematical idea in a form that's easier to visualize and work with.

• Using positive exponents supports easier interpretation and calculation.
• It allows expressions to fit standard forms in equations and functions.

In summary, converting negative exponents into positive ones is not about changing the inherent value but rather about representing it in a more conventional and workable format. This simplification is essential in solving equations and inequalities, ensuring clarity and precision in mathematical communication.