Problem 80 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{x^{n+3}}{x^{n}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression $$\frac{x^{n+3}}{x^n}$$, assuming x is nonzero and all exponents are whole numbers. Answer: $$x^3$$.

Step 1: Analyze the given problem

We are given the fraction $$\frac{x^{n+3}}{x^n}$$. First, we want to notice that we can apply the quotient rule of exponents in this problem because the bases are the same (x).

Step 2: Apply the quotient rule of exponents

In order to simplify this expression using the quotient rule, we need to subtract the exponent of the denominator from the exponent in the numerator. Here, the numerator has an exponent of $$(n+3)$$ and the denominator has an exponent of $$(n)$$. So, when applying the quotient rule, we get: $$ x^{(n+3)-n} $$

Step 3: Subtract the exponents within the parentheses

In this step, we need to subtract the exponents within the parentheses: $$ (n+3) - n = 3 $$

Step 4: Write the final simplified expression

Now that we have subtracted the exponents, we can write our final simplified expression: $$ x^3 $$ Therefore, the simplified expression is $$x^3$$.

Key Concepts

Quotient Rule of Exponents

The quotient rule of exponents is a helpful tool for simplifying expressions where the same base appears in both the numerator and the denominator of a fraction. When we talk about exponents and fractions like \(\frac{x^{a}}{x^{b}}\), the quotient rule tells us that we can simplify this by subtracting the exponent in the denominator from the exponent in the numerator.

In simpler terms:

• If you have the same base \(x\) raised to different powers like \(a\) and \(b\), you just need to subtract \(b\) from \(a\).
• The result is \(x^{a-b}\).

So, when we apply this rule to a fraction like \(\frac{x^{n+3}}{x^n}\), we subtract \(n\) from \(n+3\), which simplifies to \(x^3\). This method allows us to work with simpler expressions by reducing the complexity of the exponents.

Product Rule of Exponents

While the quotient rule is perfect for division, the product rule of exponents deals with multiplication scenarios.

With the product rule, when you multiply two expressions that have the same base, you add their exponents together. In legal terms, when you have an expression like \(x^{m} \times x^{n}\), you simply add \(m\) and \(n\) together and the result is \(x^{m+n}\).

Here's why it works:

• Multiplying numbers with the same base is like repeated addition of powers.
• This makes combining exponents a straightforward process.

This rule, like the quotient rule, helps condense expressions to more manageable forms, making calculations easier and more efficient. It complements the quotient rule when handling expressions involving multiple operations with exponents.

Simplifying Expressions

Simplifying expressions means turning a complex mathematical statement into a simpler, more digestible version. Whether you're using the quotient rule, the product rule, or other properties of exponents, the aim is to create a smaller, more convenient form.

In the given problem, we started with \(\frac{x^{n+3}}{x^n}\). By using the quotient rule, we subtracted the exponents, resulting in \(x^3\). Now, before moving to complex algebraic problems, let's focus on these simplified forms.

• This simplification helps to solve more complex equations faster.
• It also reduces the scope of errors while performing calculations.

Recognizing and using these rules effectively aids in better understanding algebraic expressions and solving them accurately. A simple expression looks friendly and is often much easier to work with, helping you face more challenging math problems with confidence.