Problem 13 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement. \(3^{2} \cdot 3^{?}=3^{7}\)

Step-by-Step Solution

Answer

The unknown exponent ? is 5.

Step 1: Identify the formula to use

The first step is recognizing the properties of exponents, specifically the rule that states that \(a^{m} \cdot a^{n} = a^{m + n}\), where a is the base and m and n are the exponents.

Step 2: Apply the formula

Next, apply the formula to the problems. This requires setting the total exponent 7 equal to the given exponent 2 plus ?, i.e. \(2 + ? = 7\).

Step 3: Solve for the unknown

The final step involves solving for the unknown ?. Simply subtract the given exponent 2 from both sides of the equation to find ? = 5.

Key Concepts

Properties of Exponents

Understanding the properties of exponents is essential for solving problems involving exponential expressions. In essence, exponents are a way to express repeated multiplication of the same number. The properties provide a set of rules that simplify computations and enable the manipulation of expressions involving exponents.One key property is the Product of Powers property. This rule states that when multiplying like bases with exponents, you add the exponents together:

• Formula: \(a^m \cdot a^n = a^{m+n}\)

Here, \(a\) represents the base number, while \(m\) and \(n\) are the exponents. This property simplifies computations because, instead of multiplying large numbers multiple times, you can just add the exponents. This can especially be helpful when working with algebraic expressions where the base remains constant. Recognizing and applying these rules can greatly reduce the complexity of exponential expressions.

Multiplication of Exponents

When you're dealing with the multiplication of exponents, it's vital to remember that the base must be the same. If the base isn't the same, you cannot simply add the exponents. Let's break it down with an example: Imagine you have \(3^2\) and \(3^5\). Since both have the base 3, you can apply the Product of Powers property. This means instead of multiplying each expression separately, you can add together the exponents:

• Formula: \(3^2 \cdot 3^5 = 3^{2+5} = 3^7\)

This method is more efficient because it reduces the steps required in calculation. The result is the same value obtained from multiplying out the factored terms by traditional methods, but with less work. This simplicity is why exponent rules are so powerful and often used for simplifying algebraic expressions.

Algebraic Expressions

In algebra, expressions are composed of numbers, variables, and operators combined into meaningful mathematical phrases. When exponents are involved, they often include complex terms that require the use of exponent rules to simplify. An algebraic expression with exponents might look like \(x^3 \cdot x^4\).To simplify expressions like this, you apply the same rules learned for numeric bases. This means, if the bases are identical, you add the exponents to reduce the expression to a simpler form:

• \(x^3 \cdot x^4 = x^{3+4} = x^7\)

By recognizing and applying these simplification rules, you make the algebraic expressions more manageable and can solve or simplify problems more efficiently. It's also important in solving equations where exponents are involved, allowing you to isolate terms and find solutions more easily.