Problem 70 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. \(12^{-5} \cdot 12^{3}\)

Step-by-Step Solution

Answer

So, the answer to the problem \(12^{-5} \cdot 12^{3}\) is \(1/144\).

Step 1: Apply the Exponent Rule

Start by applying the law of exponents for multiplication. According to this rule, \(a^{n} \cdot a^{m} = a^{n+m}\). This means that if we multiply two exponents with the same base, we add the exponents. So, our given expression \(12^{-5} \cdot 12^{3}\) becomes \(12^{(-5+3)}\).

Step 2: Simplify the Power

Now, we simplify the exponent by performing the operation of addition in the exponent, which gives us \(12^{-5+3} = 12^{-2}\).

Step 3: Calculate the Value

Next, we evaluate the expression with the power of negative two. Recall that \(a^{-n} = 1/a^n\), so we can rewrite our expression as \(1/12^2\).

Step 4: Final Calculation

Finally, calculate the value \(1/12^2 = 1/144\).

Key Concepts

Negative Exponents

Understanding negative exponents is crucial in simplifying exponential expressions. When a number has a negative exponent, it is the equivalent of taking the reciprocal of the number raised to the positive version of the exponent. In simpler terms,

• if you see something like \( a^{-n} \), it means you take \( 1 / a^n \).

This transformation helps us deal with negative powers in expressions, making them easier to work with. For example, \( 12^{-2} \) is rewritten as \( 1/12^2 \), simplifying our expression.
Negative exponents can be intimidating at first, but remembering that they denote the reciprocal operation can make them much more approachable.

Multiplication of Exponents

When working with the multiplication of exponents, it is essential to understand how to handle expressions with the same base. The key rule here is that when you multiply two exponents that have the same base, you add the exponents. For instance:

• \( a^n \cdot a^m = a^{n+m} \)

This rule simplifies the process considerably by reducing the number of steps needed to evaluate or simplify an expression.
In our original exercise, the expression \( 12^{-5} \cdot 12^{3} \) becomes \( 12^{(-5+3)} \), which further simplifies to \( 12^{-2} \). This demonstrates how addition of exponents streamlines calculations and helps in managing exponential terms effectively.

Simplifying Exponential Expressions

Simplifying exponential expressions involves systematically applying exponent rules to reduce complex expressions into simpler forms. The goal is to minimize clutter and make expressions manageable. Some essential strategies include:

• Applying the rule of adding exponents when bases are the same during multiplication.
• Rewriting negative exponents as reciprocals for ease of calculation.

For example, in simplifying \( 12^{-5} \cdot 12^{3} \), we first add \(-5+3\) to get \(12^{-2}\), then use the negative exponent rule to express it as \(1/12^2\), and finally, calculate the mathematical result.
By carefully applying these rules, we transform potentially tricky expressions into straightforward calculations, aiding in problem-solving efficiency.