Problem 55 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. (Lesson 8.1) $$3^{5} \cdot 3^{2}$$

Step-by-Step Solution

Answer

The expression \(3^{5} \cdot 3^{2}\) as a single power of the base is \(3^{7}\)

Step 1: Identify the base and the exponents

In this expression, the base is \(3\) and the exponents are \(5\) and \(2\). We want to write this as a single power of \(3\). To do this, we can add the exponents together.

Step 2: Apply the product of powers property

We apply the product of powers property \(a^{m} \cdot a^{n} = a^{m+n}\). In this case, \(a = 3\), \(m = 5\), and \(n=2\). So, it's just \(3^{5+2}\).

Step 3: Simplify the exponent

Adding the exponents together, we get \(3^{7}\)

Key Concepts

Exponent Rules

Understanding exponent rules is fundamental to working with algebraic expressions involving powers. An exponent denotes how many times a base number is multiplied by itself. For instance, when you see an expression like \(3^5\), it means \(3\) is multiplied by itself \(5\) times.

One of the basic exponent rules is the product of powers property, which states that when you multiply powers with the same base, you can simply add the exponents together. Mathematically, this is written as \(a^{m} \times a^{n} = a^{m+n}\), where \(a\) is the base, and \(m\) and \(n\) are the exponents. This rule allows for the simplification of expressions and is vital when one is working on consolidating powers or simplifying complex equations.

For example, in an expression like \(3^{5} \times 3^{2}\), we apply this rule. The base, \(3\), is the same, so we can add the exponents to get \(3^{5+2}\) or \(3^{7}\), which simplifies the expression to a single power of the base. Remember that this rule only applies when the bases are identical; if they are different, we cannot use this property to simplify the expression.

Simplifying Exponents

Simplifying exponents is another essential skill, which involves using exponent rules to make expressions more manageable. In addition to the product of powers, there are other rules such as the power of a power property and the quotient of powers property that help in simplifying exponents.

When you encounter multiple exponents, consider whether the bases are the same and which rules apply. Always look to combine terms whenever possible, according to the relevant exponent rules. For example, when simplifying \(x^{3} \times x^{4}\), we can add the exponents to get \(x^{3+4} = x^{7}\), simplifying the expression considerably.

Moreover, simplifying exponents is not only about mathematical correctness but also about clarity. In practical terms, an expression that is simplified is easier to understand and work with, especially when dealing with complex algebraic expressions. It often makes the difference between an unwieldy equation and one that is clear and concise.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent a specific quantity or equation. Variables are symbols that stand for unknown values and are often represented by letters like \(x\), \(y\), or \(z\). These expressions can range from simple forms like \(5x + 3\) to more complex ones involving exponents, such as \(3x^{3} - 2x + 7\).

The manipulation of algebraic expressions is central to algebra. It involves operations such as adding, subtracting, multiplying, and dividing, and the use of exponent rules to simplify expressions. A key part of understanding and working with these expressions is recognizing patterns and commonalities, such as the same variable or base number, which can lead to simplification through rules like the product of powers property.

In practical scenarios, algebraic expressions are used to model real-life situations, and the ability to manipulate and simplify them is crucial. Whether it’s for figuring out the growth of a bank account with compound interest or understanding a physics equation, mastering algebraic expressions and their simplification is invaluable.