Simplifying Exponential Expressions: A Step-by-Step Guide

Hey math enthusiasts! Ever stumbled upon an exponential expression and thought, "Whoa, where do I even begin?" Well, fear not! Today, we're diving deep into simplifying expressions with exponents. We'll break down the problem $(3^2 \cdot 5^4)^3$ step-by-step, making sure you grasp the concepts and can tackle similar problems with confidence. Let's get started!

Understanding the Power of a Product Rule

Before we jump into the problem, let's brush up on a crucial rule: the power of a product rule. This rule states that when you have a product raised to a power, you apply that power to each factor within the product. Mathematically, it looks like this: $(ab)^n = a^n \cdot b^n$.

Think of it like distributing the exponent. Each element inside the parentheses gets its share of the exponent's love. It's like a math party, and everyone gets to dance with the exponent! Understanding this rule is super important because it's the key to solving the main problem. So, keep this rule in mind as we move forward! It is really helpful for simplifying any expressions with exponents, that includes our question. This rule basically means that if there is a number outside the parentheses, we must multiply the number by the powers of the numbers inside the parentheses. Also, there are other exponent rules that can be useful, such as product of powers rule, power of a power rule, and quotient of powers rule. The product of powers rule states that when multiplying terms with the same base, you add the exponents. For example, $x^m \cdot x^n = x^{m+n}$. The power of a power rule states that when raising a power to another power, you multiply the exponents. For example, $(x^m)^n = x^{m \cdot n}$. The quotient of powers rule states that when dividing terms with the same base, you subtract the exponents. For example, $\frac{x^m}{x^n} = x^{m-n}$.

This principle is the cornerstone for solving our initial problem. Remember, always start by applying the power of a product rule, and then you can simplify each term further if needed. Just remember to take it step by step, and you'll be golden. The rules are designed to help you, so don't be scared to make use of them.

Solving $(3^2 \cdot 5^4)^3$: Step-by-Step

Now, let's put our knowledge to the test and break down the expression $(3^2 \cdot 5^4)^3$. We'll apply the power of a product rule, and then simplify step-by-step. First, identify what's inside the parentheses. We have $3^2$ and $5^4$. The exponent outside the parentheses is 3. So, we'll apply this exponent to each term inside. We're going to use the power of a product rule: $(ab)^n = a^n \cdot b^n$.

So, $(3^2 \cdot 5^4)^3$ becomes $3^{2 \cdot 3} \cdot 5^{4 \cdot 3}$. See how we've distributed the outer exponent to each term inside the parentheses? It's like magic! Next, we need to simplify the exponents. Multiply the exponents: $3^{2 \cdot 3}$ becomes $3^6$, and $5^{4 \cdot 3}$ becomes $5^{12}$.

Now our expression is $3^6 \cdot 5^{12}$. So, that's our simplified expression. This is the equivalent form of the original expression. Now, we just need to match it with the multiple choices given. But here, we already have the answer, and this is how easy it is to solve it. It is very important to carefully and patiently go step by step, which will help us solve the problem accurately and avoid mistakes. Always remember the fundamental rules, and practice with various examples. You'll become a pro in no time! The most common mistakes are related to the order of operations, so always pay attention to the order in which operations must be performed.

Analyzing the Answer Choices

Alright, now that we've simplified our expression to $3^6 \cdot 5^{12}$, let's see which answer choice matches. We're looking for an option that has 3 raised to the power of 6 and 5 raised to the power of 12.

Let's go through the choices:

• A. $3^2 \cdot 5^{12}$: This one is incorrect because the exponent of 3 is 2, not 6. Close, but no cigar!
• B. $3^5 \cdot 5^7$: This one is totally off. The exponents of both 3 and 5 are incorrect.
• C. $3^6 \cdot 5^{12}$: Bingo! This matches our simplified expression perfectly. The exponents of both 3 and 5 are correct.
• D. $15^{18}$: While you could technically write the original expression as $15^{18}$ (because $3^2 \cdot 5^4$ equals $9 \cdot 625 = 5625$, and $5625^3 = 15^{18}$), this isn't the simplified form we're looking for, which wants the base to be prime numbers. This is a common trick to watch out for. Always check if you need to simplify it further or match a specific form.

So, the correct answer is C. $3^6 \cdot 5^{12}$. High five!

Key Takeaways and Tips

• The Power of a Product Rule: Remember that $(ab)^n = a^n \cdot b^n$. This is your best friend when dealing with these types of problems.
• Simplify Step-by-Step: Don't rush! Break down the problem into smaller steps. This helps avoid mistakes and makes the process easier to follow.
• Know Your Exponent Rules: Review the product of powers, power of a power, and quotient of powers rules. These rules are super helpful.
• Practice Makes Perfect: The more you practice, the better you'll get. Try different examples to build your confidence.
• Pay Attention to Detail: Double-check your calculations and make sure you're applying the rules correctly.

And that's a wrap! You've successfully simplified an exponential expression. Keep practicing, and you'll be acing these problems in no time. If you have any questions, feel free to ask! Happy calculating!