Problem 2 - Elementary Algebra

Question

Find each product. $$ x^{9} \cdot x^{4} $$

Step-by-Step Solution

Answer

Answer: The product is \(x^{13}\).

Step 1: Identify the base and exponents

First, we need to identify the base and exponents of the expressions we are working with. In this case, the base is x, and the exponents are 9 and 4.

Step 2: Apply the exponent properties to multiply

Now that we have identified the base and exponents, we can apply the exponent properties to multiply the expressions together. Since we are multiplying two expressions with the same base (x), we can simply add their exponents. We have: $$ x^{9} \cdot x^{4} = x^{(9+4)} $$

Step 3: Calculate the sum of the exponents

In this step, we will calculate the sum of the exponents 9 and 4, which gives us: $$ x^{(9+4)} = x^{13} $$

Step 4: Write the final answer

To wrap up, we have found that the product of \(x^{9}\) and \(x^{4}\) is equal to \(x^{13}\). The final answer is: $$ x^{9} \cdot x^{4} = x^{13} $$

Key Concepts

Base and Exponent

The foundation of exponentiation lies in understanding the base and the exponent. The base is the number or variable that is being multiplied. The exponent tells us how many times the base is multiplied by itself.
In our example, the base is represented by "\(x\)".
When we say \(x^9\), the 9 is the exponent, which means we multiply \(x\) by itself nine times, like this:

• \(x \times x \times x \times x \times x \times x \times x \times x \times x\)

This concept helps us to simplify the expressions and perform calculations using fundamental properties of exponents.

Exponent Properties

Exponent properties are useful shortcuts that help simplify complex expressions, especially when dealing with multiplication or division.
The key property we use when multiplying terms with the same base is the "Product of Powers" property.
This property states that when you multiply two expressions with the same base, you can add their exponents. The rule is as follows:

• If you have \(a^m \cdot a^n\), it's equal to \(a^{m+n}\).

This property is what we used in our example when we combined \(x^9\) and \(x^4\) into \(x^{13}\).
This simplification makes calculations much more manageable.

Multiplication of Powers

When we multiply terms with the same base, it might look daunting at first, but it is quite straightforward due to exponent properties. As we have seen, multiplying exponents involves adding them.
Essentially, the process of multiplying powers with the same base boils down to one simple operation: addition.
Let's illustrate this with the example from our exercise:

• \(x^{9} \cdot x^{4} = x^{9+4} = x^{13}\)

This means that instead of writing the base "\(x\)" repeated 13 times individually, we use one base "\(x\)" with the summed exponent.
This is very efficient and tidies up expressions significantly, helping streamline complex algebraic expressions.