Problem 16 - Beginning and Intermediate Algebra

Question

Simplify. Assume all variables represent nonzero real numbers. The answer should not contain negative exponents. $$n^{10} \cdot n^{6}$$

Step-by-Step Solution

Answer

\(n^{16}\)

Step 1: Identify the base and exponents in the expression

In the given expression, the base is 'n' and the two exponents are 10 and 6.

Step 2: Apply the properties of exponents

When we multiply two terms with the same base, we add their exponents. In this case, we have \(n^{10} \cdot n^{6}=n^{(10+6)}\).

Step 3: Simplify the exponent

Now, we need to add the exponents 10 and 6, which results in 16. So our simplified expression becomes: \(n^{(10+6)}=n^{16}\).

Step 4: Ensure no negative exponents

The final simplified expression is \(n^{16}\), which doesn't contain any negative exponents. Therefore, our simplification is complete.

Key Concepts

Properties of Exponents

Exponents have several important properties that make manipulating them easier. These properties help us simplify expressions like the one given. Let's take a look at some of the key properties you should know:

• **Product of Powers Property**: When multiplying two exponents with the same base, you can add the exponents together. For example, \(a^m \cdot a^n = a^{m+n}\).
• **Power of a Power Property**: Raising a power to another power multiplies the exponents. For example, \((a^m)^n = a^{m \cdot n}\).
• **Power of a Product Property**: This states that the exponent applies to each factor in the product separately. For example, \((ab)^n = a^n \cdot b^n\).

Understanding these key properties can significantly ease the process of simplifying exponential expressions. They provide a structured way to handle operations that might otherwise seem complex.

Simplifying Exponents

Simplifying exponents means writing expressions in their simplest form, so they are easier to understand and use in calculations. In our example, we dealt with the expression \(n^{10} \cdot n^{6}\). Since the bases are the same, we can apply the Product of Powers Property.

By adding the exponents, we get:

• \(n^{10+6} = n^{16}\)

This approach ensures that our expression remains efficient and straightforward. It's important to follow the rule of combining exponents correctly to avoid mistakes in simplification. Also, always check to ensure that no negative exponents are present after simplifying unless specified, as the simplified result in most cases should involve only positive exponents.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operation symbols that represent a value or set of values. Knowing how to transform and simplify these expressions is crucial for problem-solving in math.

In the context of exponents, a mathematical expression might include terms with bases raised to various powers. The task could involve simplifying or solving these expressions, as seen with \(n^{10} \cdot n^{6}\). Each expression is designed to follow specific operation rules like addition, multiplication, and especially properties of exponents when variables exhibit powers. Simplifying these expressions step-by-step, using the properties of exponents, helps achieve clarity and consistency, which is invaluable in both academic and real-world mathematical applications.

Furthermore, once we master these expressions, we can easily evaluate and manipulate more complex scenarios or compound expressions in algebra and beyond.