Simplifying Exponential Expressions A Step-by-Step Guide

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Hey guys! Ever stumbled upon an exponential expression that looks like a mathematical maze? Well, today we're going to break down one such expression step by step, making it super easy to understand. We'll be diving deep into how to simplify ${\frac{3^{x+9}+3^{x+1}}{2^{x+3}-2^{x+1}}}$ so you can confidently tackle similar problems. Let's get started!

Understanding the Basics of Exponential Expressions

Before we jump into the nitty-gritty, let's quickly recap what exponential expressions are all about. At its core, an exponential expression involves a base raised to a power, also known as an exponent. For example, in ${3^2}$, 3 is the base, and 2 is the exponent. The exponent tells us how many times to multiply the base by itself. So, ${3^2}$ means 3 multiplied by itself, which equals 9.

Now, when these exponential terms start appearing in fractions or more complex expressions, things can seem a bit daunting. But don't worry! The trick is to use the properties of exponents to simplify them. These properties are like our secret weapons in this mathematical adventure. We'll be using properties like the product of powers, quotient of powers, and the power of a power. These rules help us to manipulate and simplify exponential expressions effectively.

Let's briefly touch on some key properties that we'll use in our simplification process:

1. Product of Powers: When you multiply two exponents with the same base, you add the exponents. Mathematically, this is represented as ${a^m \times a^n = a^{m+n}}$.
2. Quotient of Powers: When you divide two exponents with the same base, you subtract the exponents. This is expressed as ${\frac{a^m}{a^n} = a^{m-n}}$.
3. Power of a Power: When you raise a power to another power, you multiply the exponents. This rule is written as ${(a^m)^n = a^{mn}}$.

These properties might seem like abstract rules now, but you'll see how powerful they are as we start applying them to our expression. Remember, the goal is to break down the complex problem into smaller, manageable parts. By understanding these exponential properties, we lay the foundation for simplifying even the most intimidating expressions. So, keep these rules in mind, and let's move on to our main challenge: simplifying ${\frac{3^{x+9}+3^{x+1}}{2^{x+3}-2^{x+1}}}$!

Breaking Down the Numerator: Simplifying ${3^{x+9}+3^{x+1}}$

Okay, guys, let's tackle the numerator of our expression: ${3^{x+9}+3^{x+1}}$. The key here is to use the properties of exponents to factor out a common term. This is like finding a common thread that ties the two terms together, making them easier to handle.

Our main focus here is identifying a common exponential factor. Notice that both terms have a base of 3, and the exponents involve ${x}$. The lowest exponent involving ${x}$ is ${x+1}$. So, we can rewrite the terms to factor out ${3^{x+1}}$.

Let's break it down:

• ${3^{x+9}}$ can be rewritten as ${3^{(x+1)+8}}$. Using the product of powers rule, we can further express this as ${3^{x+1} \times 3^8}$. Remember, when you multiply exponential terms with the same base, you add the exponents. So, ${3^{x+1} \times 3^8}$ is the same as ${3^{(x+1)+8}}$, which simplifies back to ${3^{x+9}}$.
• The second term, ${3^{x+1}}$, is already in a simple form, so we don't need to change it.

Now, we can rewrite the numerator as:

${3^{x+9}+3^{x+1} = 3^{x+1} \times 3^8 + 3^{x+1}}$

See how ${3^{x+1}}$ appears in both terms? This is our common factor! We can factor it out, just like you would factor out a common number in a regular algebraic expression. This is a crucial step in simplifying the numerator. Factoring out the common term makes the expression much cleaner and easier to work with.

So, let's factor out ${3^{x+1}}$:

${3^{x+1} \times 3^8 + 3^{x+1} = 3^{x+1}(3^8 + 1)}$

By factoring out the common term, we have transformed the sum into a product. This is a significant simplification because it allows us to potentially cancel out terms later on, especially when we consider the denominator. Also, it helps to condense the expression, making it less cumbersome.

Now, let's simplify the expression inside the parentheses. We know that ${3^8}$ means 3 multiplied by itself 8 times. Calculating this gives us 6561. So, we can replace ${3^8}$ with 6561:

${3^{x+1}(3^8 + 1) = 3^{x+1}(6561 + 1)}$

Adding 1 to 6561 gives us 6562. So, our simplified numerator is:

${3^{x+1}(6562)}$

Wow! Look how much simpler the numerator has become. We started with ${3^{x+9}+3^{x+1}}$ and, through careful factoring and simplification, we've arrived at ${3^{x+1}(6562)}$. This is a huge step forward in simplifying the entire expression. Next up, we'll tackle the denominator using a similar approach. Keep up the great work, guys!

Simplifying the Denominator: Cracking ${2^{x+3}-2^{x+1}}$

Alright, team, let's shift our focus to the denominator: ${2^{x+3}-2^{x+1}}$. Just like we did with the numerator, our goal here is to simplify this expression by identifying and factoring out a common term. This will help us to reduce the complexity and make the entire fraction easier to manage.

Looking at the two terms, ${2^{x+3}}$ and ${2^{x+1}}$, we can see that they both have a base of 2 and exponents involving ${x}$. The strategy remains the same: find the term with the lowest power of 2 involving ${x}$ and factor it out. In this case, the lowest exponent is ${x+1}$, so we'll aim to factor out ${2^{x+1}}$.

Let's break down the process step by step:

• Consider the term ${2^{x+3}}$. We can rewrite this using the properties of exponents. Specifically, we can express ${x+3}$ as ${(x+1)+2}$. So, ${2^{x+3}}$ becomes ${2^{(x+1)+2}}$. Using the product of powers rule, we can rewrite this as ${2^{x+1} \times 2^2}$. Remember, this rule states that ${a^{m+n} = a^m \times a^n}$. So, ${2^{(x+1)+2}}$ is the same as ${2^{x+1} \times 2^2}$.
• The second term, ${2^{x+1}}$, is already in its simplest form with respect to the common factor we're looking for. We don't need to change it.

Now, we can rewrite the denominator as:

${2^{x+3}-2^{x+1} = 2^{x+1} \times 2^2 - 2^{x+1}}$

Notice that ${2^{x+1}}$ is a common factor in both terms. Factoring out this common term will simplify the expression significantly. It's like pulling out the common thread in a mathematical fabric, making the whole structure more manageable.

So, let's factor out ${2^{x+1}}$:

${2^{x+1} \times 2^2 - 2^{x+1} = 2^{x+1}(2^2 - 1)}$

By factoring out the common term, we've transformed the difference into a product. This is a crucial step towards simplifying the entire expression. It allows us to potentially cancel out terms later on, which is always a satisfying moment in mathematical simplification.

Next, let's simplify the expression inside the parentheses. We know that ${2^2}$ is equal to 4. So, we can replace ${2^2}$ with 4:

${2^{x+1}(2^2 - 1) = 2^{x+1}(4 - 1)}$

Subtracting 1 from 4 gives us 3. Thus, our simplified denominator is:

${2^{x+1}(3)}$

Fantastic! We've successfully simplified the denominator from ${2^{x+3}-2^{x+1}}$ to ${2^{x+1}(3)}$. This is a significant simplification that brings us closer to our final answer. We've broken down the denominator, making it much easier to work with. Now, we're ready to put the simplified numerator and denominator together and see if we can simplify the entire fraction further.

Putting It All Together: Simplifying the Entire Fraction

Okay, everyone, we've done the hard work of simplifying both the numerator and the denominator separately. Now comes the exciting part: putting it all together and seeing how much further we can simplify the entire fraction. This is where we'll see the fruits of our labor, and it's always a satisfying moment in problem-solving.

Let's recap what we've found so far:

• We simplified the numerator, ${3^{x+9}+3^{x+1}}$, to ${3^{x+1}(6562)}$.
• We simplified the denominator, ${2^{x+3}-2^{x+1}}$, to ${2^{x+1}(3)}$.

Now, let's write the entire fraction with our simplified expressions:

${\frac{3^{x+9}+3^{x+1}}{2^{x+3}-2^{x+1}} = \frac{3^{x+1}(6562)}{2^{x+1}(3)}}$

Here's where the magic happens! Look closely at the fraction. Do you see any common factors that we can cancel out? The presence of common factors is what makes the entire simplification process worthwhile. Spotting these common elements can significantly reduce the complexity of the expression.

We can rewrite 6562 as a product of its prime factors to see if there are any common factors with 3. Prime factorization of 6562 gives us ${2 \times 3281}$. So, we can rewrite the fraction as:

${\frac{3^{x+1}(2 \times 3281)}{2^{x+1}(3)}}$

Unfortunately, there isn't a direct cancellation of exponential terms here, but we can simplify the numerical part of the fraction. Notice that 6562 has a factor of 2, and the denominator has a factor of 3. We can separate the fraction into numerical and exponential parts to make it clearer:

${\frac{3^{x+1}(6562)}{2^{x+1}(3)} = \frac{3^{x+1}}{2^{x+1}} \times \frac{6562}{3}}$

Now, let's simplify the numerical fraction ${\frac{6562}{3}}$. Dividing 6562 by 3 gives us approximately 2187.33, which isn't a whole number. However, we made a mistake in the prime factorization of 6562. The correct prime factorization of 6562 is ${2 \times 17 \times 193}$. This means that 6562 does not have 3 as a factor. So, we cannot simplify the fraction ${\frac{6562}{3}}$ further.

Let’s express the exponential part as a single term. We can rewrite ${\frac{3^{x+1}}{2^{x+1}}}$ using the property ${\frac{a^n}{b^n} = (\frac{a}{b})^n}$. Applying this rule, we get:

${\frac{3^{x+1}}{2^{x+1}} = (\frac{3}{2})^{x+1}}$

So, our expression becomes:

${(\frac{3}{2})^{x+1} \times \frac{6562}{3}}$

This is as simplified as we can get the expression without further information or context. We've combined the exponential terms and simplified the numerical fraction as much as possible. We've successfully navigated the complexities of the original expression and arrived at a more manageable form. Great job, everyone!

Final Simplified Form and Conclusion

Alright, team, we've reached the end of our mathematical journey for today! We started with a complex-looking exponential expression, (\frac{3^{x+9}+3{x+1}}{2^{x+3}-2{x+1}}, and through a series of strategic simplifications, we've arrived at a much cleaner and understandable form. This is what mathematical problem-solving is all about: breaking down complex problems into manageable parts and applying the right tools and techniques to solve them.

Let's recap our journey step by step:

1. Understanding the Basics: We started by revisiting the fundamentals of exponential expressions and the key properties of exponents that would guide our simplification process. This foundational knowledge was crucial for tackling the problem effectively.
2. Simplifying the Numerator: We focused on the numerator, ${3^{x+9}+3^{x+1}}$, and identified the common factor ${3^{x+1}}$. Factoring out this term allowed us to rewrite the numerator as ${3^{x+1}(6562)}$. This step was a significant simplification, making the expression much easier to handle.
3. Simplifying the Denominator: Next, we turned our attention to the denominator, ${2^{x+3}-2^{x+1}}$. Similar to the numerator, we identified the common factor ${2^{x+1}}$ and factored it out, simplifying the denominator to ${2^{x+1}(3)}$. This step was equally important in reducing the complexity of the original expression.
4. Putting It All Together: With both the numerator and the denominator simplified, we combined them into a single fraction: ${\frac{3^{x+1}(6562)}{2^{x+1}(3)}}$. We then looked for opportunities to cancel out common factors, but the numerical part required careful consideration.
5. Final Touches: We rewrote the fraction as a product of the exponential part and the numerical part: ${(\frac{3}{2})^{x+1} \times \frac{6562}{3}}$. We recognized that ${\frac{6562}{3}}$ could not be simplified further as 6562 is not divisible by 3. This led us to our final simplified form.

So, after all the hard work, our final simplified expression is:

${(\frac{3}{2})^{x+1} \times \frac{6562}{3}}$

This is a much more manageable form than our original expression. It clearly shows the exponential relationship and the numerical factor. Simplifying expressions like this is not just a mathematical exercise; it's a skill that's valuable in many areas of science and engineering.

In conclusion, by understanding the properties of exponents and applying strategic factoring techniques, we were able to simplify a complex exponential expression. Remember, the key to success in mathematics is to break down problems into smaller, more manageable steps. Keep practicing, and you'll become a pro at simplifying even the most challenging expressions. Great job today, guys! You've nailed it!