Expressing Expressions As Powers: A Step-by-Step Guide

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Hey guys! Let's dive into the exciting world of exponents and learn how to express various expressions as powers. This is a fundamental concept in mathematics, and mastering it will definitely help you in more advanced topics. So, grab your pencils, and let's get started!

Understanding the Basics of Exponents

Before we jump into the problems, let's quickly recap what exponents are all about. An exponent, or power, indicates how many times a base number is multiplied by itself. For example, in the expression 2^3 (read as "2 to the power of 3"), 2 is the base, and 3 is the exponent. This means we multiply 2 by itself three times: 2 * 2 * 2 = 8. Understanding this basic principle is crucial for tackling the problems ahead.

  • The base is the number being multiplied. 2 is the number being multiplied in 2^3.
  • The exponent is the number of times the base is multiplied by itself. In 2^3, 3 is the exponent.

When we work with exponents, some key rules come into play. The most important one for our exercises today is the product of powers rule. This rule states that when multiplying powers with the same base, you can add the exponents together. Mathematically, it's expressed as: a^m * a^n = a^(m+n). This rule simplifies complex expressions into more manageable forms.

For instance, if we have 2^2 * 2^3, we can use this rule. Both terms have the same base (2), so we add the exponents (2 and 3) to get 2^(2+3) = 2^5. Therefore, 2^2 * 2^3 is equal to 2^5, which is 32. This rule is a cornerstone in simplifying expressions involving exponents, saving us from having to perform individual multiplications.

It's also important to remember that any number raised to the power of 0 is equal to 1 (a^0 = 1). This might seem counterintuitive at first, but it's a fundamental rule in exponent manipulation. Additionally, any number raised to the power of 1 is simply the number itself (a^1 = a). These are the foundational blocks we'll be using to solve the exercises, making sure we accurately combine and simplify different exponential forms.

Solving the Expressions: Step-by-Step

Now, let's tackle the expressions you provided. We'll break down each one step-by-step, so you can clearly see how we apply the rules of exponents.

a) 2^2 * 2^3 * 2^0 * 2

In this expression, we have multiple terms with the same base (2) being multiplied together. To express this as a single power, we need to add the exponents. Remember that when a number is written without an exponent, it's implicitly raised to the power of 1. So, 2 is the same as 2^1.

Following the product of powers rule, we add all the exponents:

2 + 3 + 0 + 1 = 6

Therefore, 2^2 * 2^3 * 2^0 * 2 can be expressed as 2^6. It's as simple as adding up the powers when the bases are the same. This transformation makes the expression far more concise and easier to work with in further calculations.

b) 5^3 * 5^5 * 5 * 5^10

This expression is similar to the previous one, but with a different base (5). Again, we have multiple terms with the same base being multiplied. We'll apply the same rule: add the exponents. Remember that 5 is the same as 5^1.

Adding the exponents:

3 + 5 + 1 + 10 = 19

So, 5^3 * 5^5 * 5 * 5^10 can be written as 5^19. This highlights the power of the product rule, efficiently combining terms into a single, simplified exponential representation.

c) 27 * 3 * …

Okay, this one's a bit tricky because it's incomplete! To express this as a power, we need to figure out what the "…" represents. Let's assume the question intends to have only 27 * 3. In such a case, the first step is to express both numbers with the same base. We know that 27 is 3^3. So, we can rewrite the expression as:

3^3 * 3

Now, we can apply the product of powers rule. Remember, 3 is the same as 3^1.

Adding the exponents:

3 + 1 = 4

Therefore, 27 * 3 can be expressed as 3^4. If there are more terms in the original expression, we would need those to provide a complete solution.

d) 4^2 * 8^3 * 2^5 * 16

This expression looks a bit more complex because the bases are different (4, 8, 2, and 16). To apply the product of powers rule, we need to express all the terms with the same base. The smallest base here is 2, so let's convert everything to base 2.

  • 4 can be written as 2^2
  • 8 can be written as 2^3
  • 16 can be written as 2^4

Now, substitute these into the original expression:

(22)2 * (23)3 * 2^5 * 2^4

We have another rule of exponents to apply here: the power of a power rule, which states (am)n = a^(m*n). This means we multiply the exponents when raising a power to another power.

Applying this rule:

2^(22) * 2^(33) * 2^5 * 2^4

2^4 * 2^9 * 2^5 * 2^4

Now that all terms have the same base, we can add the exponents:

4 + 9 + 5 + 4 = 22

Therefore, 4^2 * 8^3 * 2^5 * 16 can be expressed as 2^22. Converting all bases to a common base before applying the product of powers rule is the key strategy here.

e) 47 * 47^2 * 47^5

This expression is straightforward. All terms have the same base (47). Remember that 47 is the same as 47^1. Now, we just add the exponents:

1 + 2 + 5 = 8

So, 47 * 47^2 * 47^5 can be written as 47^8. It’s a simple application of the product rule, emphasizing the efficiency gained when the bases are the same.

f) 5^0 * 5^1

This one's another simple application of the product of powers rule. We add the exponents:

0 + 1 = 1

Therefore, 5^0 * 5^1 can be expressed as 5^1, which is simply 5. Remember that any number raised to the power of 0 is 1, so this could also be seen as 1 * 5^1 = 5.

Key Takeaways and Tips

So, guys, expressing expressions as powers involves a few key steps and rules. Let's recap the most important ones:

  1. Identify the base: Make sure all terms have the same base before applying the product of powers rule.
  2. Product of powers rule: When multiplying powers with the same base, add the exponents (a^m * a^n = a^(m+n)).
  3. Power of a power rule: When raising a power to another power, multiply the exponents ((am)n = a^(m*n)).
  4. Convert to the same base: If the bases are different, try to express all terms with the smallest base.
  5. Remember the basics: Any number raised to the power of 0 is 1 (a^0 = 1), and any number raised to the power of 1 is itself (a^1 = a).

By mastering these rules and practicing regularly, you'll become a pro at simplifying exponential expressions. Keep up the great work, and remember, math can be fun!

Practice is key. Try more problems, and you'll soon become comfortable with manipulating exponents. Don't hesitate to review the rules whenever you get stuck. Remember, every mathematician was once a beginner!

If you found this guide helpful, share it with your friends and classmates. Let's conquer math together! And if you have any more questions, feel free to ask. Keep exploring, keep learning, and have fun with math!Strong understanding of exponents not only aids in solving equations but also builds a solid foundation for more advanced mathematical concepts. So, keep practicing and keep learning. You've got this!