Problem 83 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression. $$ \frac{7^{4} \cdot 7}{7^{7}} $$

Step-by-Step Solution

Answer

So, the simplified form of the expression is \(7^{-2}\).

Step 1: Understand exponentiation rules

The first step implies recognizing the rule \(a^{m} \cdot a^{n} = a^{m+n}\). This rule allows us to add exponents when the bases are the same.

Step 2: Apply the rule

Apply the rule to the given expression. The base '7' is the same for all parts of the equation. This means we can add the exponent '4' and '1' (representing \(7^{4} \cdot 7\)) to get the exponent '5'. So the expression becomes \(7^{5}/7^{7}\).

Step 3: Simplify the expression

Finally, use the rule \(a^{m}/a^{n} = a^{m-n}\) to subtract the exponent in the denominator from the exponent in the numerator. So the expression becomes \(7^{5-7} = 7^{-2}\).

Key Concepts

Exponentiation Rules

Exponentiation is a way to express repeated multiplication of the same number. Understanding exponentiation rules is crucial for simplifying expressions effectively. There are several key rules to remember:

• Product of Powers Rule: This states that when multiplying two expressions that have the same base, you can add the exponents: \(a^{m} \cdot a^{n} = a^{m+n}\).
• Quotient of Powers Rule: This involves dividing two expressions with the same base. Here, you subtract the exponents: \(a^{m}/a^{n} = a^{m-n}\).

By using these rules, you can simplify complex expressions into simpler terms, making calculations much easier. Getting comfortable with these foundational rules will help you tackle more challenging problems with confidence.

Same Base Exponents

When simplifying expressions, it's helpful to identify exponents with the same base. This enables you to use exponentiation rules effectively. For example, if you have multiple terms like \(7^4 \) and \(7\), both share the base of \(7\).
This commonality allows you to apply the product of powers rule, combining them into one term with a single base. Recognizing the same base:

• Simplifies expressions.
• Reduces the number of steps needed.
• Helps visualize the expression better.

Being able to spot same-base exponents quickly is an essential skill in algebra. It streamlines the problem-solving process and aids in maintaining accuracy.

Subtraction of Exponents

Subtraction of exponents occurs during division when the bases are the same. This applies the quotient of powers rule, where you subtract the exponent in the denominator from the exponent in the numerator. For example, given \(\frac{7^5}{7^7}\), you perform \(5 - 7\) to get \(7^{-2}\).
This simple rule helps in:

• Reducing complex fractions.
• Turning expressions into a more manageable form.
• Easily identifying negative exponents, which indicate \(\frac{1}{a^n}\).

Understanding subtraction in terms of exponents demystifies division within exponential expressions. Practice applying this rule to develop an intuitive grasp of how it operates in various scenarios.