Problem 60 - Elementary Algebra

Question

Simplify \(x^{n+3} \cdot x^{5}\).

Step-by-Step Solution

Answer

Question: Simplify the expression \(x^{n+3} \cdot x^5\). Answer: \(x^{n+8}\).

Step 1: Identify the base and exponents

In this expression, our base is \(x\), and the exponents are \((n+3)\) and \(5\). Our goal is to simplify the expression by combining the exponents using the properties of exponents.

Step 2: Apply the exponent product rule

To do this, we can use the product rule for exponents: \(x^{a} \cdot x^{b} = x^{a+b}\) In our exercise, \(a = n+3\) and \(b = 5\). Applying the product rule, we combine the exponents: \(x^{(n+3)} \cdot x^{5} = x^{(n+3) + 5}\)

Step 3: Simplify the exponent

Now, we just have to simplify the exponent: \((n+3) + 5 = n + 8\) So, we can write the simplified expression as: \(x^{n+8}\)

Step 4: Check the final answer

Our final simplified expression is \(x^{n+8}\), so the answer is: \(x^{n+3} \cdot x^{5} = x^{n+8}\).

Key Concepts

Exponent Product Rule

Understanding the exponent product rule is essential for simplifying expressions with exponents. This rule states that when you multiply two algebraic expressions with the same base, you can add the exponents. Essentially, it's like saying 'if I have a certain number of groups of the same size, I can simply count all the items together instead of counting each group separately.'

For example, using the rule \(x^{a} \times x^{b} = x^{a+b}\), we can simplify the expression \(x^{n+3} \times x^{5}\) to \(x^{n+3+5}\), which is then further simplified to \(x^{n+8}\). It's important to note that this rule applies only when the bases of the exponents are the same, as similar terms can be combined.

In practice, always look for opportunities to use this rule when you encounter terms that can be combined. It's a powerful tool that can often make complex problems much simpler.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations that represent a specific value or range of values. In the expression \(x^{n+3} \times x^{5}\), we observe two main components: the variable \(x\), which could represent any number, and the exponents, which tell us how many times to multiply \(x\) by itself.

A crucial part of working with algebraic expressions is understanding how to manipulate them. This can involve things like combining like terms, factoring, expanding, and simplifying expressions with exponents. The ability to work fluently with algebraic expressions is foundational for higher mathematics, such as calculus and beyond.

When simplifying algebraic expressions, always ensure each step is clear, use parentheses to avoid confusion, and perform operations in the correct order, keeping the 'order of operations' (PEMDAS/BODMAS) in mind.

Properties of Exponents

Properties of exponents provide us with rules for handling expressions that involve powers. Beyond the product rule are other properties such as the quotient rule \(x^{a} \div x^{b} = x^{a-b}\), the power of a power rule \(\(x^{a}\)^{b} = x^{ab}\), and the zero exponent rule \(x^{0} = 1\), among others.

Understanding these properties is vital for simplifying expressions and solving equations involving exponents. These rules are consistent for all real numbers and can greatly simplify the process of manipulating exponent expressions, making even the most daunting equations manageable.

To improve work with properties of exponents, one should practice using these rules in many different contexts to become confident in recognizing when and how to apply them. The more familiar one becomes with these properties, the easier it will be to tackle algebraic problems that appear complex at first glance.