Problem 14 - Beginning and Intermediate Algebra

Question

Simplify. $$n^{9} \cdot n^{-9}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(n^{9} \cdot n^{-9} = 1\).

Step 1: Identify the exponent rule to be applied

We will apply the multiplication property of exponents, which states that \(a^{m} \cdot a^{n} = a^{m+n}\).

Step 2: Apply the multiplication property of exponents

By applying the multiplication property of exponents, we will add the exponents of \(n\), as follows: \(n^{9} \cdot n^{-9} = n^{9 + (-9)}\)

Step 3: Calculate the result

Now we need to compute the result of the addition inside the exponent: \(n^{9 + (-9)} = n^{0}\)

Step 4: Apply the rule about any base raised to the power of zero

According to the exponent rules, any base raised to the power of zero is equal to 1: \(n^{0} = 1\) So, the simplified form of the given expression is: \(n^{9} \cdot n^{-9} = 1\)

Key Concepts

Multiplication Property of Exponents

The multiplication property of exponents is a fundamental rule in algebra that helps simplify expressions with similar bases. When multiplying two powers with the same base, you add their exponents. This can be expressed by the formula:

• \(a^{m} \cdot a^{n} = a^{m+n}\)

This property makes calculations easier by combining the powers into a single term. For example, for the expression \(n^{9} \cdot n^{-9}\), we simply add the exponents:

• \(9 + (-9) = 0\)

So it reduces to \(n^{0}\). This process is often confusing at first, but with practice, it becomes an intuitive part of working with algebraic expressions.

Zero Exponent Rule

The zero exponent rule is another essential concept in algebra. According to this rule, any non-zero number raised to the power of zero is equal to 1. This can be startling initially, but it derives from the properties of exponents themselves.

• If \(a eq 0\), then \(a^{0} = 1\).

This rule helps simplify many algebraic expressions, especially when dealing with the multiplication of terms that result in a zero exponent, as in our example:

• \(n^{0} = 1\)

Understanding why this rule works involves considering how dividing powers with the same base works. This simplification greatly reduces complexity in algebraic expressions and is crucial for learning more advanced mathematics concepts.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form by applying various rules and properties of exponents. It often includes combining like terms, handling coefficients, and leveraging exponent properties.

• Use the multiplication property to combine terms.
• Apply the zero exponent rule when applicable.

The goal is to make expressions as simple as possible without changing their value. With practice, these rules help you quickly and accurately streamline complicated expressions, like turning \(n^{9} \cdot n^{-9}\) into \(1\). Incorporating these rules in your algebra toolkit makes problem-solving more efficient and less error-prone, laying a strong foundation for solving more complex problems.

Algebra Basics

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is the foundational language of mathematics and involves learning to work within a system of rules involving exponents, variables, and operations.

• Understand the properties of numbers and their operations.
• Learn to manipulate symbols to solve equations.

Basic concepts include understanding terms, expressions, and equations. Mastery of these basics allows students to progress to more advanced topics, such as linear equations, functions, and polynomials. The given problem urges the use of exponent rules, which is a vital skill in simplifying algebraic expressions and is key to succeeding in more advanced mathematical studies.