Simplify & Evaluate Exponent Expressions: A Step-by-Step Guide

by ADMIN 63 views
Iklan Headers

Hey guys! Today, we're diving deep into the fascinating world of exponents and how we can use their properties to simplify complex expressions. We'll be tackling a specific problem, breaking it down step-by-step, and making sure you understand the logic behind each move. So, buckle up and get ready to become exponent experts!

The Challenge: Rewriting and Evaluating an Expression

Our mission, should we choose to accept it (and we do!), is to tackle this expression:

(2a4)2a5a5\frac{\left(2 a^4\right)^2}{a^5 a^5}

We've got some exponents in the numerator, some in the denominator, and even a coefficient thrown in for good measure. Our goal is twofold: first, we need to rewrite this expression in its simplest form using the properties of exponents. Then, we need to evaluate that simplified expression when $a = -5$. Sounds like a plan, right?

Step 1: Mastering the Properties of Exponents

Before we jump into the problem, let's quickly review the key exponent properties that we'll be using. These are the bread and butter of simplifying expressions with exponents, so make sure you've got them down!

  • Product of Powers: When multiplying exponents with the same base, we add the powers. Mathematically, this looks like: $x^m * x^n = x^{m+n}$
  • Power of a Power: When raising a power to another power, we multiply the exponents. The formula here is: $(xm)n = x^{m*n}$
  • Power of a Product: When raising a product to a power, we distribute the power to each factor within the product. This can be written as: $(xy)^n = x^n * y^n$
  • Quotient of Powers: When dividing exponents with the same base, we subtract the powers. This property is expressed as: $\frac{xm}{xn} = x^{m-n}$

With these properties in our arsenal, we're ready to take on the expression!

Step 2: Simplifying the Numerator

Let's focus on the numerator first: $(2a4)2$. We need to apply the power of a product property here. This means we distribute the exponent of 2 to both the coefficient (2) and the variable term ($a^4$).

Applying the power of a product rule, we get:

(2a4)2=22∗(a4)2(2a^4)^2 = 2^2 * (a^4)^2

Now, we simplify $2^2$ to 4. And for $(a4)2$, we use the power of a power property, multiplying the exponents 4 and 2:

22∗(a4)2=4∗a4∗2=4a82^2 * (a^4)^2 = 4 * a^{4*2} = 4a^8

So, the simplified numerator is $4a^8$. We've conquered the first hurdle!

Step 3: Simplifying the Denominator

Next up, let's tackle the denominator: $a^5 a^5$. This is a straightforward application of the product of powers property. We have the same base ('a') and we're multiplying, so we add the exponents:

a5a5=a5+5=a10a^5 a^5 = a^{5+5} = a^{10}

Excellent! The simplified denominator is $a^{10}$. We're making great progress, guys!

Step 4: Combining the Simplified Numerator and Denominator

Now that we've simplified both the numerator and the denominator, let's put them back together in our fraction:

(2a4)2a5a5=4a8a10\frac{(2 a^4)^2}{a^5 a^5} = \frac{4a^8}{a^{10}}

This is looking much cleaner already! We're ready for the final simplification step.

Step 5: Applying the Quotient of Powers Property

We now have a fraction with the same base ('a') in both the numerator and the denominator. This is where the quotient of powers property comes into play. We subtract the exponent in the denominator from the exponent in the numerator:

4a8a10=4a8−10=4a−2\frac{4a^8}{a^{10}} = 4a^{8-10} = 4a^{-2}

We've got a negative exponent! While this is a simplified form, it's often preferred to express exponents with positive values. To do this, we use the property that $x^{-n} = \frac{1}{x^n}$. So, we rewrite $a^{-2}$ as $\frac{1}{a^2}$:

4a−2=4∗1a2=4a24a^{-2} = 4 * \frac{1}{a^2} = \frac{4}{a^2}

Boom! We've successfully rewritten the expression in its simplest form: $\frac{4}{a^2}$. Give yourselves a pat on the back!

Step 6: Evaluating the Expression When a = -5

Now for the second part of our mission: evaluating the simplified expression when $a = -5$. This is the fun part where we get to plug in a value and see what we get.

We substitute -5 for 'a' in our simplified expression:

4a2=4(−5)2\frac{4}{a^2} = \frac{4}{(-5)^2}

Remember that when squaring a negative number, the result is positive. So, $(-5)^2 = (-5) * (-5) = 25$

Now we have:

4(−5)2=425\frac{4}{(-5)^2} = \frac{4}{25}

And there you have it! The value of the rewritten expression when $a = -5$ is $\frac{4}{25}$.

The Answer and Why

Looking back at the options provided (A. -20, B. -80, C. -250, D. -500), none of them match our calculated value of $\frac{4}{25}$. This means either there was a mistake in the original options, or the question intended to have $ rac{4}{25}$ as a possible answer.

It's incredibly important to double-check your work and the provided options in these types of problems. Sometimes, mistakes happen, and it's crucial to be confident in your solution! Our step-by-step approach has led us to the correct simplified expression and its value, so we can be sure of our answer.

Key Takeaways: Exponents are Your Friends!

Guys, we've covered a lot in this guide! We've not only solved a challenging problem but also reinforced the fundamental properties of exponents. Remember these key takeaways:

  • Master the properties: The product of powers, power of a power, power of a product, and quotient of powers are your best friends when simplifying expressions with exponents.
  • Break it down: Complex problems become manageable when you break them down into smaller, logical steps.
  • Double-check everything: Always review your work and the provided options to ensure accuracy. Math is like a puzzle, and every piece needs to fit perfectly!
  • Positive Exponents are Preferred: While negative exponents are mathematically correct, it is generally preferred to express the final answer using positive exponents.

By following these strategies, you'll be able to confidently tackle any exponent problem that comes your way. Keep practicing, and you'll become an exponent master in no time!

Practice Makes Perfect: Try These Problems!

To solidify your understanding, try simplifying and evaluating these expressions using the same techniques we covered:

  1. (3b2)3b4b\frac{(3b^2)^3}{b^4 b}

  2. 5x5(x2)2\frac{5x^5}{ (x^2)^2}

  3. (4y3)2∗y−2(4y^3)^2 * y^{-2}

Work through these problems, and don't hesitate to review the steps we discussed in this guide. With a little practice, you'll be simplifying exponent expressions like a pro! Happy calculating, friends!

Conclusion: Exponent Experts Unite!

We've reached the end of our exponent adventure! Remember, guys, that mastering exponents is a crucial step in your mathematical journey. By understanding and applying the properties we've discussed, you'll be well-equipped to tackle more advanced concepts in algebra and beyond. Keep practicing, stay curious, and never stop exploring the amazing world of mathematics! You got this!