Mastering Exponents: Step-by-Step Solutions

Hey guys! Let's dive into calculating exponential results. This guide will break down each problem step-by-step, making it super easy to understand. We'll cover everything from basic exponents to more complex expressions. So, grab your calculators (or your brains!) and let's get started!

a. 5² 𝑋 5⁶

When dealing with exponents, especially when the bases are the same, things get pretty straightforward. In this case, we have 5² multiplied by 5⁶. The golden rule here is: when multiplying exponential expressions with the same base, you simply add the exponents. So, what does this look like in practice?

We start with 5² 𝑋 5⁶. According to the rule, we add the exponents 2 and 6. This gives us 2 + 6 = 8. Therefore, the expression simplifies to 5⁸. Now, to find the actual result, we need to calculate 5 raised to the power of 8. This means multiplying 5 by itself eight times: 5 𝑋 5 𝑋 5 𝑋 5 𝑋 5 𝑋 5 𝑋 5 𝑋 5.

Calculating this, we find that 5⁸ equals 390,625. So, the final answer to the expression 5² 𝑋 5⁶ is 390,625. This principle is super useful in simplifying complex calculations and is a fundamental concept in algebra. Whether you're dealing with scientific notation or solving algebraic equations, understanding how to manipulate exponents is crucial. Remember, the key is to keep the base the same and simply add the exponents when multiplying. Easy peasy!

Understanding exponents is like having a superpower in math. You can quickly simplify what looks like a complicated expression into something manageable. So, keep practicing, and soon you'll be a pro at handling exponents!

b. 12¹⁰ / 12⁷

Moving on to division, we encounter the expression 12¹⁰ divided by 12⁷. Just like with multiplication, there's a neat rule to follow when dividing exponential expressions with the same base: you subtract the exponents. So, instead of adding, we're subtracting. Cool, right?

Starting with 12¹⁰ / 12⁷, we subtract the exponents 7 from 10. This gives us 10 - 7 = 3. Thus, the expression simplifies to 12³. To find the actual value, we calculate 12 raised to the power of 3, which means multiplying 12 by itself three times: 12 𝑋 12 𝑋 12.

When we do the math, we find that 12³ equals 1,728. Therefore, the final result of the expression 12¹⁰ / 12⁷ is 1,728. This rule is super handy because it makes simplifying complex divisions much easier. Imagine trying to calculate 12¹⁰ and 12⁷ separately and then dividing – that would take forever! By simply subtracting the exponents, we save a ton of time and reduce the chance of making errors.

Mastering these exponent rules can significantly boost your confidence in handling algebraic problems. Keep in mind that these rules are consistent and reliable, making them an invaluable tool in your mathematical toolkit. So, keep practicing, and you'll find that these types of problems become second nature. Remember, math is all about understanding the rules and applying them correctly!

c. 8² 𝑋 8⁴

Now, let's tackle another multiplication problem: 8² multiplied by 8⁴. Just like before, we're dealing with the same base (which is 8), so we can apply the rule of adding the exponents. This makes things super simple and quick. Who doesn’t love a shortcut, am I right?

We start with 8² 𝑋 8⁴. We add the exponents 2 and 4, which gives us 2 + 4 = 6. Therefore, the expression simplifies to 8⁶. Now, let's calculate 8 raised to the power of 6. This means multiplying 8 by itself six times: 8 𝑋 8 𝑋 8 𝑋 8 𝑋 8 𝑋 8.

After crunching the numbers, we find that 8⁶ equals 262,144. So, the final answer to the expression 8² 𝑋 8⁴ is 262,144. Understanding this rule is essential because it helps you simplify expressions quickly without having to calculate large exponents manually. This is particularly useful in scenarios where you're dealing with very large numbers or in algebraic contexts where simplifying expressions is crucial.

The key takeaway here is that when you see the same base being multiplied with different exponents, remember to add those exponents. It's a simple rule that can save you a lot of time and effort. Keep practicing, and you'll become super efficient at solving these types of problems. Math is all about finding the easiest and most effective way to get to the solution!

d. (9³)³

Let's move on to a slightly different type of problem: (9³)³. This involves an exponent raised to another exponent. When this happens, we have another rule: you multiply the exponents. So, instead of adding or subtracting, we're multiplying. Ready to see how it works?

We start with (9³)³. According to the rule, we multiply the exponents 3 and 3. This gives us 3 𝑋 3 = 9. Therefore, the expression simplifies to 9⁹. Now, to find the actual result, we need to calculate 9 raised to the power of 9. This means multiplying 9 by itself nine times: 9 𝑋 9 𝑋 9 𝑋 9 𝑋 9 𝑋 9 𝑋 9 𝑋 9 𝑋 9.

Calculating this, we find that 9⁹ equals 387,420,489. So, the final answer to the expression (9³)³ is 387,420,489. This rule is especially useful when dealing with nested exponents, as it simplifies the expression into a single exponent, making it much easier to manage. It’s like a mathematical superpower for simplifying complex expressions!

Understanding and applying this rule can significantly reduce the complexity of many algebraic problems. Remember, when you have an exponent raised to another exponent, you simply multiply them. Keep practicing, and you'll find these problems becoming much more straightforward. Math is all about recognizing patterns and applying the right rules!

e. (1/4)⁵ 𝑋 (1/4)⁶

Finally, let's tackle an expression involving fractions: (1/4)⁵ multiplied by (1/4)⁶. Even though we're dealing with fractions, the same exponent rules apply. Since the bases are the same (1/4), we can simply add the exponents. Don't let the fraction scare you; it’s easier than it looks!

We start with (1/4)⁵ 𝑋 (1/4)⁶. We add the exponents 5 and 6, which gives us 5 + 6 = 11. Therefore, the expression simplifies to (1/4)¹¹. Now, to find the actual result, we need to calculate 1/4 raised to the power of 11. This means multiplying 1/4 by itself eleven times: (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4) 𝑋 (1/4).

Calculating this, we find that (1/4)¹¹ equals approximately 0.0000002384. So, the final answer to the expression (1/4)⁵ 𝑋 (1/4)⁶ is approximately 0.0000002384. This demonstrates that the rules of exponents apply regardless of whether the base is a whole number or a fraction. This is super useful because it means you don't have to learn different rules for different types of numbers!

The key takeaway here is that the exponent rules are universal. They work for whole numbers, fractions, and even variables. Keep practicing with different types of bases, and you'll become a master of exponents in no time. Remember, math is all about consistency and applying the same rules in different contexts!

So there you have it! Calculating exponential results can be a breeze once you understand and apply the basic rules. Keep practicing, and you'll be solving these problems like a pro. Happy calculating, guys!