Problem 1 - Elementary Algebra

Question

Find each product. $$ x^{2} \cdot x^{5} $$

Step-by-Step Solution

Answer

Question: Find the product of the expressions $$x^{2}$$ and $$x^{5}$$. Answer: The product of the expressions is $$x^{7}$$.

Step 1: Identify the rule

We will use the product rule of exponents, which states: $$ a^{m} \cdot a^{n} = a^{m+n} $$ In our case, both expressions have the base x, so we can apply this rule.

Step 2: Apply the rule

Now we will apply the product rule of exponents to our given expressions: $$ x^{2} \cdot x^{5} = x^{(2+5)} $$

Step 3: Simplify and find the product

Let's simplify the expression by adding the exponents: $$ x^{(2+5)} = x^{7} $$ Our final answer is: $$ x^{2} \cdot x^{5} = x^{7} $$

Key Concepts

Product Rule of Exponents

The product rule of exponents is a fundamental concept in algebra that simplifies expressions involving powers of the same base. This rule is quite straightforward and highly useful when working with exponential expressions. When you have two exponents with the same base, such as \(a^m \cdot a^n\), you can combine them using the product rule. This rule states that you simply add the exponents together while keeping the base the same.An example is \(x^2 \cdot x^5\). According to the product rule, you add the exponents: \(2 + 5 = 7\), so the expression simplifies to \(x^7\). This makes the rule handy when dealing with algebraic expressions, as it reduces complexity by reducing the number of terms in an expression. Remember, this only works if the base is the same; different bases can't be combined using this rule.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations. These expressions can include terms involving exponents, and understanding how to manipulate these expressions is crucial in algebra.Let's break it down:

• **Variables** represent unknown values, often noted by letters like \(x\), \(y\), or \(z\).
• **Constants** are fixed numbers with known values, like 5 or -3.
• **Operators** such as addition (+), subtraction (-), and multiplication (\(\cdot\)) combine terms.

For example, in the expression \(x^2 \cdot x^5\), \(x\) is the variable, 2 and 5 are the exponents, and \(\cdot\) is the operator indicating multiplication.Handling these expressions often relies on algebraic rules, like the product rule of exponents, to simplify and solve them efficiently. By rewriting complex expressions into simpler forms, algebraic expressions become much easier to work with.

Simplifying Expressions

Simplifying expressions is all about rewriting them in the most reduced form possible, making them easier to understand and work with. In algebra, this process often involves combining like terms, using arithmetic operations, and applying exponent rules.

Why Simplify?

Simplifying helps to solve problems faster and ensures solutions are presented in the clearest form. It's particularly helpful when you're dealing with complex equations or expressions, allowing you to see potential solutions or problems with ease.

Steps to Simplify Exponential Expressions

• **Identify the Parts:** Look for terms with the same base and exponents.
• **Apply Rules:** Use properties such as the product rule of exponents to combine terms.
• **Combine and Reduce:** Add or subtract exponents as needed, ensuring the base remains unchanged.

In our example \(x^2 \cdot x^5\), we simplified the expression by adding exponents: \(2 + 5 = 7\), resulting in \(x^7\). This makes working with the expression more straightforward. Simplified expressions are not only easier to handle in further calculations but also essential for accurately solving more complex algebraic problems.