Problem 4 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \frac{7^{6}}{7^{9}} $$

Step-by-Step Solution

Answer

The simplified expression is \(7^{-3}\).

Step 1: Identify the base and the exponents

Identify the base, which is 7 in this case, and the exponents in the numerator and the denominator. Here the exponent in the numerator is 6 and in the denominator is 9.

Step 2: Apply the quotient of powers property

Using the quotient of powers property, subtract the exponent of the denominator from the exponent of the numerator. The expression then simplifies to \(7^{6-9}\).

Step 3: Perform the subtraction

Subtract the exponents: 6 - 9 equals -3. So, the simplified expression is \(7^{-3}\).

Key Concepts

Simplifying Expressions with Exponents

Expressions with exponents can often appear intimidating, but simplifying them is a straightforward process when you know the rules. One common situation is when we need to divide two powers with the same base. Imagine you're working with large numbers of identical factors divided by each other—it's like having a large group of friends and then noticing some of them leave the party, so you have fewer friends left. Similarly, when dividing exponents with the same base, you're essentially cancelling out a certain number of these repeated factors.

For example, when you see \(\frac{a^{m}}{a^{n}}\), you can think of the base \(a\) being multiplied by itself \(m\) times in the numerator, and \(n\) times in the denominator. Using the quotient of powers property, we simplify the expression by subtracting the exponent in the denominator from the exponent in the numerator to get \(a^{m-n}\). This rule reflects the process of removing the common factors as we're effectively reducing the repeated multiplication. It aims to make an intimidating string of factors much more manageable, leading to a simplified expression that is easier to understand and work with.

Exponent Subtraction

Exponent subtraction is a principle that lays the foundation for simplifying expressions with exponents. The core of this concept is found within the quotient of powers property, which tells us that to divide two powers with the same base, you subtract the exponents. It's as if you're keeping score in a game: if your team scores \(m\) points and the opposing team scores \(n\) points, to find the difference, you subtract the smaller number from the larger one. Likewise, with exponents, subtracting them tells you how many of the base's factors remain.

Take the expression \(\frac{7^{6}}{7^{9}}\) as an example. By subtracting the exponent of the denominator from the numerator (6 - 9), we find a new exponent, which is -3, simplifying our expression to \(7^{-3}\). This simplified form tells us that we have three fewer sevens on top than on the bottom—an essential step in simplifying expressions to their most basic form.

Negative Exponents

Let's tackle the often-misunderstood world of negative exponents. These are not the nefarious entities they might seem at first sight. In fact, they follow a simple rule: a negative exponent indicates that the base is on the 'wrong' side of the fraction line and needs to be inverted. So when you come across an expression like \(7^{-3}\), this doesn't mean that 7 is now a negative number. Instead, it conveys that we should 'flip' the base to the other side of the fraction.

In more technical terms, \(a^{-n}\) is the same as \(\frac{1}{a^{n}}\). So for our example, \(7^{-3}\) would become \(\frac{1}{7^{3}}\), or \(\frac{1}{343}\) when simplified further. Understanding how negative exponents flip the base of the fraction from numerator to denominator helps demystify expressions that might otherwise seem more complex than they truly are. It's like having a map that reveals a shortcut to your destination—the destination being a simpler and more elegant expression.