Problem 12 - Beginning and Intermediate Algebra

Question

Simplify. $$6^{2} \cdot 6^{-7}$$

Step-by-Step Solution

Answer

The simplified expression is \(6^{-5}\).

Step 1: Identify the Product Rule for exponents

With the same base, when multiplying, the exponents can be added. In this case, the base is 6 and the exponents are 2 and -7.

Step 2: Apply the Product Rule

Following the product rule, add the exponents: \[ 6^2 \cdot 6^{-7} = 6^{(2+(-7))} \]

Step 3: Simplify the expression

Now, compute the addition inside the parentheses: \[ 6^{(2+(-7))} = 6^{(-5)} \] So, the simplified expression is: \[ 6^2 \cdot 6^{-7} = 6^{-5} \]

Key Concepts

Product Rule

The product rule for exponents is an essential tool when you're dealing with expressions that have the same base. In simple terms, it tells us that when we multiply powers with the same base, we can add their exponents.
This makes calculations much simpler. For instance, if you have an expression like \(a^m \cdot a^n\), you can simplify it to \(a^{m+n}\). The base "a" remains the same, while the exponents "m" and "n" are added together.
This rule is incredibly useful in quickly simplifying complex expressions and is based on the principle that multiplication is a form of repeated addition.

Simplifying Expressions

Simplifying expressions involving exponents often relies on understanding a few basic rules, such as the product rule, which we touched on earlier.
By using these rules, you can reduce an expression to its simplest form, making it easier to work with or to solve further problems. For example, in the expression \(6^2 \cdot 6^{-7}\), we can use the product rule to combine the exponents into a single power:

• First, identify the base (which is the same for both powers, here it's 6)
• Next, add the exponents (2 and -7) together
• This results in the new exponent: \(2 + (-7) = -5\)

Now, the expression is simplified to \(6^{-5}\). Keeping expressions simple can help you solve equations more efficiently and avoid errors along the way.

Negative Exponents

Negative exponents can be a bit confusing at first, but they simply represent the reciprocal of a number raised to a positive exponent.
When you see an expression with a negative exponent, such as \(a^{-n}\), it's equivalent to \(\frac{1}{a^n}\). This means the number is flipped "upside down" or inverted.
In the simplified expression \(6^{-5}\), we can convert this into a fraction:

• The base 6 becomes the denominator
• The exponent 5 is applied to this base
• So, \(6^{-5} = \frac{1}{6^5}\)

This conversion makes it easier to understand and manipulate the expression. Recognizing that negative exponents indicate reciprocals is key to mastering exponent rules.