Calculate Expressions: Step-by-Step Math Guide

Hey guys! Today, we're diving into some cool math problems involving exponents and basic arithmetic. These problems might look a bit intimidating at first, but don’t worry, we'll break them down step by step so you can easily understand them. We're going to tackle two expressions that involve exponents, division, subtraction, and addition. Let's get started!

Part A: Solving [(3²)⁸ : 3¹³ - 3² - 3⁰] * 8 : 3⁴ + 10²

In this first part, we'll solve the expression [(3²)⁸ : 3¹³ - 3² - 3⁰] * 8 : 3⁴ + 10². This looks complicated, but we'll take it one step at a time. Always remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction.

First off, let's get into those parentheses! Our focus is on simplifying what’s inside the brackets: (3²)⁸ : 3¹³ - 3² - 3⁰. To get started, we need to remember the rules of exponents. When you have a power raised to another power, like (3²)⁸, you multiply the exponents. So, (3²)⁸ becomes 3^(2*8) which is 3¹⁶. Now our expression inside the brackets looks like this: 3¹⁶ : 3¹³ - 3² - 3⁰.

Next up, we have division involving exponents. When you divide numbers with the same base, you subtract the exponents. So, 3¹⁶ : 3¹³ becomes 3^(16-13) which simplifies to 3³. Now we're dealing with: 3³ - 3² - 3⁰. Let's calculate these exponential terms. 3³ is 3 * 3 * 3, which equals 27. 3² is 3 * 3, giving us 9. And remember, any number raised to the power of 0 is 1, so 3⁰ is 1. Now we rewrite the expression: 27 - 9 - 1.

Alright, let's handle the subtraction! First, 27 minus 9 is 18. Then, 18 minus 1 is 17. So, the entire expression inside the brackets simplifies to 17. That’s a big chunk done!

Now, let’s bring the rest of the equation into play: 17 * 8 : 3⁴ + 10². We tackle multiplication and division from left to right. 17 multiplied by 8 is 136. So, we now have: 136 : 3⁴ + 10². We need to calculate 3⁴, which is 3 * 3 * 3 * 3, and that's 81. Our expression now looks like this: 136 : 81 + 10².

Division time! 136 divided by 81 is a bit tricky, and it doesn't result in a whole number. We can leave it as a fraction for now: 136/81. Moving on, let’s deal with 10². This is 10 * 10, which equals 100. So, our expression is now: 136/81 + 100. To add these together, we need a common denominator. We'll convert 100 to a fraction with a denominator of 81. 100 is the same as 100/1, so we multiply both the numerator and denominator by 81 to get 8100/81. Now we have: 136/81 + 8100/81.

Now we can add the fractions! 136 plus 8100 is 8236. So, we have 8236/81. This is an improper fraction, meaning the numerator is larger than the denominator. You can leave it like this, or convert it to a mixed number by dividing 8236 by 81. The result is approximately 101.68. So, the final answer for part a is 8236/81 or approximately 101.68. Great job, guys! That was a tough one, but we made it through together. Remember, breaking it down into smaller steps is key!

Part B: Solving (11¹² : 11⁸ + 5³ - 4⁴ * 4⁸) : [(11²)² + 5³ - (4⁴)³]

Now, let's jump into the second part of our problem: (11¹² : 11⁸ + 5³ - 4⁴ * 4⁸) : [(11²)² + 5³ - (4⁴)³]. This one looks like a beast, but don't worry! We'll apply the same principles we used before: PEMDAS/BODMAS and exponent rules. We're going to tackle each set of parentheses separately, then deal with the division.

Let's start with the first set of parentheses: (11¹² : 11⁸ + 5³ - 4⁴ * 4⁸). First up, we have division with the same base: 11¹² : 11⁸. Remember, we subtract the exponents: 11^(12-8) = 11⁴. Next, we have 5³, which means 5 * 5 * 5 = 125. Now, let's tackle the multiplication part: 4⁴ * 4⁸. When multiplying with the same base, we add the exponents: 4^(4+8) = 4¹². So, our expression now looks like this: 11⁴ + 125 - 4¹².

Let's calculate these exponents. 11⁴ is 11 * 11 * 11 * 11 = 14641. 4¹² is a much larger number: 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 * 4 = 16777216. Now our expression is: 14641 + 125 - 16777216. Let’s do the addition first: 14641 + 125 = 14766. Now we subtract: 14766 - 16777216 = -16762450. So, the first set of parentheses simplifies to -16762450.

Time to move on to the second set of parentheses: [(11²)² + 5³ - (4⁴)³]. Again, we start with the exponents. (11²)² means we multiply the exponents: 11^(22) = 11⁴. 5³ we already know is 125. Now, (4⁴)³ means we multiply the exponents: 4^(43) = 4¹². So, the expression inside the second set of brackets becomes: 11⁴ + 125 - 4¹².

Hey, wait a minute! This looks familiar. We already calculated these values in the first part! 11⁴ is 14641, and 4¹² is 16777216. So, this expression is the same as before: 14641 + 125 - 16777216. Adding 14641 and 125 gives us 14766. Subtracting 16777216 from 14766 results in -16762450. So, the second set of parentheses also simplifies to -16762450.

Now we have our simplified expression: -16762450 : -16762450. Anything divided by itself is 1! So, the final answer for part b is 1. Woohoo! We conquered it!

Key Takeaways

• Order of Operations: Always follow PEMDAS/BODMAS. Parentheses/Brackets first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
• Exponent Rules: Remember, when you have a power raised to a power, you multiply the exponents. When you multiply numbers with the same base, you add the exponents. When you divide numbers with the same base, you subtract the exponents.
• Break It Down: Complex problems become much easier when you break them into smaller, manageable steps.

Final Thoughts

These kinds of problems are great for sharpening your math skills! By understanding the basic rules and taking things one step at a time, you can tackle even the most intimidating expressions. Keep practicing, and you'll become a math whiz in no time! Thanks for joining me today, guys. Keep up the awesome work, and I'll catch you in the next one! Remember, math can be fun when you approach it with the right attitude and strategies.