Problem 32 - Algebra and Trigonometry

Question

\(29-46\) Simplify each expression. $$ x^{-5} x^{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are given the expression \(x^{-5} x^{3}\) and we need to simplify it using the properties of exponents.

Step 2: Apply the Power of a Product Rule

According to the exponent rule, when you multiply two powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\). Here, we apply this rule to simplify \(x^{-5} \cdot x^{3}\).

Step 3: Simplification of Exponents

Add the exponents of \(x\): \(-5 + 3 = -2\). Therefore, \(x^{-5} x^{3} = x^{-2}\).

Step 4: Simplify Negative Exponent

A negative exponent indicates a reciprocal. Therefore, \(x^{-2} = \frac{1}{x^2}\). This is the simplified form of the expression.

Key Concepts

Negative Exponent

Exponents are not just about increasing numbers. They also signify division when they are negative. A negative exponent tells us to take the reciprocal of the base raised to the positive of that exponent. For example, if we have a base of \(x\) with a negative exponent, such as \(x^{-2}\), this can be rewritten as \(\frac{1}{x^2}\). This means we are inverting the base and multiplying it by itself as many times as the positive exponent indicates.

• This operation is crucial in algebra because it helps us alter the expression from potentially complex forms to simpler, more manageable ones.
• Negative exponents, therefore, do not inherently provide negative outcomes but instead, represent division.

It's important to understand this concept well, as it is fundamental when dealing with different exponent rules and simplifying expressions.

Product of Powers

The 'product of powers' property is a handy rule when working with exponents. It comes into play when you multiply two expressions that have the same base. The rule states that you can add the exponents. In mathematical terms, if you have \(a^m \cdot a^n\), you can simplify this to \(a^{m+n}\).
Applying this rule can make complex multiplication much simpler. For the expression \(x^{-5} \cdot x^3\), we identify that both have the same base \(x\). Hence, we add the exponents: \(-5 + 3 = -2\). Therefore, the simplified version of the expression is \(x^{-2}\).

• This property is essential because it allows you to combine like terms easily without directly multiplying large numbers.
• It's a time-saving strategy, especially useful in algebra and calculus.

Simplifying Expressions

Simplifying expressions is a process that makes mathematical expressions more accessible and easier to work with. When simplifying an expression with multiple exponents, we can use the rules of exponents to combine and reduce them.
For example, to simplify \(x^{-5} \cdot x^3\), we follow these steps:

• Identify common bases, here \(x\), and then apply the product of powers rule, which gives us \(x^{-2}\).
• Next, apply the negative exponent rule, thus translating \(x^{-2}\) to \(\frac{1}{x^2}\).
• This expresses the final simplified form, which is easier to interpret and solve further problems with, if necessary.

Simplifying makes expressions cleaner, which helps in accurately evaluating and manipulating them within larger problems.