Problem 50 - Algebra 1: Concepts and Skills

Question

Copy and complete the statement using \(<\) or \(>\). \(7^{3} \cdot 7^{4} \quad ? \quad(7 \cdot 7)^{4}\)

Step-by-Step Solution

Answer

\(7^{3} \cdot 7^{4} < (7 \cdot 7)^{4}\)

Step 1: Simplify expressions with the same base

On the left side, you have \(7^{3} \cdot 7^{4}\). According to the rule of multiplying powers with the same base, you add the exponents. Hence, you get \(7^{3+4} = 7^{7}\).

Step 2: Simplify power of a power

On the right side, you have \((7 \cdot 7)^{4}\). According to the rule of power of a power, you multiply the exponents. Hence, you get \((7^{2})^{4} = 7^{2 \cdot 4} = 7^{8}\).

Step 3: Compare the expressions

Now this leaves us with \(7^{7}\) and \(7^{8}\). As the bases are the same, the comparison is hinged on the exponent. In this case, 7 is less than 8. Therefore, \(7^{7} < 7^{8}\).

Key Concepts

Multiplying Powers with the Same Base

When we encounter multiplication involving exponents with the same base, there's a simple rule to follow: we just add the exponents. This is known as the Product of Powers Property. It's a critical concept in algebra because it simplifies calculations and provides a method for working with exponential expressions efficiently.

For instance, let's look at an example, such as multiplying two powers of 7: \(7^3 \cdot 7^4\). We keep the base the same (\(7\) in this case) and simply add the exponents: \(3\) and \(4\). This gives us: \(7^{3+4} = 7^7\). Easy, right?

Remember, this rule only applies when the bases are the same. If they are different, this rule cannot be used, and other methods must be applied to simplify the expression.

Power of a Power

Moving onto another exponent rule, when we raise a power to another power, we multiply the exponents. This is known as the Power of a Power Property. It's a powerful tool (pun intended!) for simplifying expressions with stacked exponents.

Taking our textbook example \((7 \cdot 7)^4\), or more generally, \((b^m)^n\), the rule tells us to multiply the exponents \(m\) and \(n\) together. For our number 7: \((7^2)^4 = 7^{2 \cdot 4} = 7^8\). This property helps emphasize the importance of order in operations—first, we consider the exponent within the parentheses, and then we deal with the exponent outside.

Comparing Exponents

Now that we understand the basics of multiplying powers and the power of a power, we naturally progress to comparing exponents. This is all about figuring out which of two exponential expressions is larger when they have either a common base or exponent.

Returning to our textbook case, we are comparing \(7^7\) and \(7^8\). Since the base (\(7\)) is the same, the expression with the larger exponent (\(8\) over \(7\)) is the greater one. Hence, \(7^7 < 7^8\). This comparison process is straightforward as long as the bases or exponents are the same, introducing an element of simplicity to potential complexity in exponential expressions.

Patience and practice are key here. As you work with more examples, these concepts will become second nature, simplifying complex problems and paving the way to mastery of algebraic exponents.