Simplifying Exponential Expressions: A Step-by-Step Guide

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Hey guys! Math can sometimes look like a monster, especially when you're staring down a complex expression filled with exponents and fractions. But don't worry, we're going to break down a seemingly complicated problem into super manageable steps. In this guide, we'll tackle the simplification of the expression (((5a2)/2.6)(-3)) . So, grab your pencils, and let's get started!

Understanding the Initial Expression

Let's take a closer look at what we're dealing with. The expression (((5a2)/2.6)(-3)) might seem intimidating at first glance, but it’s just a combination of a fraction raised to a negative power. To effectively simplify this, we need to recall a few key principles of exponents. The main idea here is to handle the negative exponent and then deal with the fraction inside the parentheses. Breaking it down like this makes the whole process way less scary.

  • Negative Exponents: Remember, a negative exponent means we need to take the reciprocal of the base. In simpler terms, x^(-n) = 1/(x^n). This is a fundamental rule that we will use to flip the fraction inside the parentheses.
  • Fractional Base: We have a fraction inside the parentheses (((5a^2)/2.6)). To raise this fraction to a power, we need to raise both the numerator (the top part) and the denominator (the bottom part) to that power. Think of it like distributing the exponent to each part of the fraction.
  • Power of a Product: When we have a product raised to a power, like (ab)^n, we raise each factor to that power: a^n * b^n. This will be important when we deal with the 5a^2 term.

By keeping these principles in mind, we can approach this problem with confidence and simplify it step by step. No need to rush, guys – understanding each step is the key here.

Step 1: Dealing with the Negative Exponent

The first thing we need to tackle is that pesky negative exponent. Remember, a negative exponent tells us to take the reciprocal of the base. So, (((5a2)/2.6)(-3)) becomes (((2.6)/(5a2))3). See? We just flipped the fraction inside the parentheses, and now our exponent is positive. This is a crucial step because it transforms the expression into a more manageable form.

Now, let's break down why this works. When you have something raised to the power of -1, you're essentially doing 1 divided by that thing. So, x^(-1) is the same as 1/x. When you have a negative exponent other than -1, like -3 in our case, it means you're doing 1 divided by that thing raised to the positive exponent. So, x^(-3) is the same as 1/(x^3). In our expression, (((5a2)/2.6)(-3)) means 1/(((5a2)/2.6)3). To get rid of the complex fraction, we can multiply the numerator (which is 1) by the reciprocal of the denominator, which gives us (((2.6)/(5a2))3).

Flipping the fraction simplifies the entire expression and sets us up for the next steps. Remember, the goal is to make the expression easier to work with, and getting rid of the negative exponent is a big win.

Step 2: Applying the Positive Exponent

Okay, we've flipped the fraction, and our expression now looks like (((2.6)/(5a2))3). The next step is to apply the exponent of 3 to both the numerator and the denominator. This means we need to raise both 2.6 and 5a^2 to the power of 3. Think of it like distributing the exponent across the fraction.

So, we get (2.6^3) / ((5a2)3). Now, let's break down what this means:

  • Numerator: 2.6^3 means 2.6 multiplied by itself three times: 2.6 * 2.6 * 2.6. This gives us 17.576. So, the numerator becomes 17.576.
  • Denominator: Here, we have (5a2)3. This means we need to raise both 5 and a^2 to the power of 3. Remember the power of a product rule? (ab)^n = a^n * b^n. So, (5a2)3 becomes 5^3 * (a2)3.
    • 5^3 is 5 multiplied by itself three times: 5 * 5 * 5, which equals 125.
    • (a2)3 is a power raised to another power. In this case, we multiply the exponents: a^(23) = a^6*. This is another important exponent rule to remember: (xm)n = x^(mn)*.

Putting it all together, the denominator becomes 125a^6. So, our expression now looks like 17.576 / (125a^6). We're making great progress, guys!

Step 3: Simplifying the Expression

We've reached a point where the expression looks much simpler: 17.576 / (125a^6). Now, let's see if we can simplify it further. The first thing we can do is look at the numerical part of the fraction, which is 17.576 / 125. To make this easier to work with, we can perform the division.

  • Dividing 17.576 by 125, we get approximately 0.140608. So, our expression becomes 0.140608 / a^6.

Now, let's think about what this means. We have a decimal number divided by a variable raised to a power. In this case, there isn't much more we can do to simplify the expression further in terms of basic arithmetic. The a^6 in the denominator tells us that the variable 'a' is raised to the sixth power, and there are no like terms to combine or cancel out.

So, our simplified expression is 0.140608 / a^6. This is a pretty clean and concise form of the original expression. Sometimes, the simplest form is a decimal divided by a variable term, and that's perfectly okay!

Step 4: Double-Checking and Final Answer

Before we proudly declare our victory, it’s always a good idea to double-check our work. Let’s quickly recap the steps we took:

  1. We started with (((5a2)/2.6)(-3)).
  2. We dealt with the negative exponent by flipping the fraction: (((2.6)/(5a2))3).
  3. We applied the exponent to both the numerator and the denominator: (2.6^3) / ((5a2)3), which became 17.576 / (125a^6).
  4. Finally, we simplified the numerical part: 0.140608 / a^6.

Looking back at each step, we can see that we've applied the rules of exponents correctly and simplified the expression logically. We handled the negative exponent, distributed the positive exponent, and simplified the resulting fraction. Everything looks good!

Therefore, the simplified form of (((5a2)/2.6)(-3)) is 0.140608 / a^6. That's our final answer!

Alternative Representation (Fraction Form)

While the decimal form 0.140608 / a^6 is perfectly acceptable, sometimes it's helpful to represent the numerical coefficient as a fraction. This can make the expression look even cleaner and might be preferred in certain contexts.

To convert the decimal 0.140608 to a fraction, we can write it as 140608/1000000. Now, we need to simplify this fraction by finding the greatest common divisor (GCD) of 140608 and 1000000 and dividing both the numerator and the denominator by it. This might seem a bit tedious, but there are tools and methods to help us find the GCD.

Alternatively, we can work backwards from our earlier step, 17.576 / 125. We know that 17.576 came from 2.6^3. So, we can rewrite 17.576 as 2.6 * 2.6 * 2.6. To get rid of the decimal, we can multiply each 2.6 by 10, which means we'll also need to multiply the denominator by 10^3 (since we multiplied the numerator by 10 three times).

So, we have (26 * 26 * 26) / (125 * 1000), which simplifies to 17576 / 125000. Now, we can simplify this fraction by dividing both the numerator and the denominator by their GCD. Both numbers are divisible by 8, so let's divide by 8: (17576 / 8) / (125000 / 8) = 2197 / 15625.

Therefore, the fraction form of our expression is (2197 / 15625) / a^6, or we can write it as 2197 / (15625a^6). This is an equally valid and often preferred way to represent the simplified expression.

Key Takeaways

So, what did we learn today, guys? Simplifying exponential expressions might seem tough at first, but by breaking it down into steps and remembering the key rules of exponents, it becomes much more manageable. Let's recap the key takeaways from this exercise:

  • Negative Exponents: A negative exponent means taking the reciprocal of the base (x^(-n) = 1/(x^n)).
  • Power of a Fraction: Apply the exponent to both the numerator and the denominator ((a/b)^n = a^n / b^n).
  • Power of a Product: Raise each factor to the power ((ab)^n = a^n * b^n).
  • Power of a Power: Multiply the exponents ((xm)n = x^(mn)).
  • Simplifying Fractions: Look for common factors to reduce the fraction to its simplest form.

By mastering these rules and practicing regularly, you'll be able to tackle even the most intimidating exponential expressions with confidence. Keep practicing, and you'll become a math whiz in no time!

Practice Problems

Now that we've walked through this example together, why not try some practice problems to solidify your understanding? Here are a few for you to try:

  1. Simplify (((3b3)/4)(-2)).
  2. Simplify (((2x4)/1.5)(-4)).
  3. Simplify (((7c2)/3.5)(-3)).

Work through these problems step by step, just like we did in the example. Remember to deal with the negative exponent first, then apply the positive exponent, and finally simplify the expression. Don't be afraid to make mistakes – that's how we learn! The key is to practice, practice, practice.

Conclusion

Well, guys, we've reached the end of our simplification journey! We took a complex exponential expression and broke it down into manageable steps, applied the rules of exponents, and arrived at a simplified answer. Whether you prefer the decimal form or the fraction form, you now have the tools to tackle similar problems with confidence.

Remember, math isn't about memorizing formulas; it's about understanding the underlying principles and applying them logically. So, keep practicing, keep exploring, and never stop asking questions. You've got this!