Problem 54 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. (Lesson 8.1) $$7^{6} \cdot 7^{2}$$

Step-by-Step Solution

Answer

The expression \(7^{6} \cdot 7^{2}\) written as a single power of the base is \(7^{8}\)

Step 1: Identify the Base and the Powers

In the given expression \(7^{6} \cdot 7^{2}\), the base is 7 and the powers are 6 and 2, respectively.

Step 2: Apply Exponent Addition Rule

Use the exponent rule that states the multiplication of two exponents with the same base is equal to the base raised to the sum of the two powers, i.e, \(a^{n} \cdot a^{m} = a^{n+m}\). Here, adding the powers 6 and 2 gives \(7^{6+2} = 7^{8}\).

Step 3: Final Expression

The simplified form of \(7^{6} \cdot 7^{2}\) is \(7^{8}\)

Key Concepts

Power of a Base

When dealing with exponents, the term 'power of a base' refers to the expression of a number like \(b^n\), where \(b\) is the base and \(n\) is the exponent. This indicates that the base \(b\) is multiplied by itself \(n\) times. For instance, in the expression \(7^6\), 7 is the base, and 6 is the power, meaning \(7\) is multiplied by itself 6 times. Understanding the base and its power is crucial because it allows us to simplify and manipulate expressions when they appear in mathematical problems. Consider them as building blocks. The base stays the same while you can adjust the power depending on the situation, especially when multiplying bases with common exponential rules. This simplification is a key skill that helps in various mathematical operations.

Simplifying Expressions

Simplifying expressions involves the process of making complex mathematical expressions easier to understand and work with. When you come across expressions like \(7^6 \cdot 7^2\), you aim to combine them into a more straightforward form by using exponent rules. To simplify, you first identify similar bases. Here, both powers have the base 7. The next step is to apply relevant exponent rules to transform this expression into a simpler format.A simple form makes it easier to understand the quantity represented and allows for more efficient calculations in further mathematical procedures. By simplifying expressions with the same bases, you save time and reduce the possibility of errors, especially in long and intricate calculations.

Addition of Exponents

In the world of exponents, one of the fundamental rules is the addition of exponents when you're multiplying like bases. This rule states that for any non-zero base \(a\), \(a^n \cdot a^m = a^{n+m}\). What this means is, when you multiply expressions with the same base, you can simply add their exponents. For example, in the exercise \(7^6 \cdot 7^2\), both exponents share the base 7. According to our rule, you add 6 and 2, resulting in \(7^{6+2} = 7^8\).This rule is pivotal in mathematics as it greatly simplifies the process of working with large powers. By replacing multiple smaller calculations with a single power expression, you'll not only save time but also keep your work organized and clear. Understanding and applying this rule effectively allows you to tackle more advanced problems with confidence.