Solving Algebraic Expressions: A Step-by-Step Guide

Hey guys! Let's dive into some algebra and figure out how to solve these expressions step by step. We'll break down each problem, so you can easily understand the process. Don't worry, it's not as scary as it sounds! We'll start with the basics and move on to more complex ones. Ready to get started? Let's go!

a) $9^2 + 19$

Finding the value of expressions can seem daunting at first, but it's really about following the order of operations and doing one step at a time. This first problem is a great example of that. Here, we have $9^2 + 19$. Remember, the order of operations (often remembered by the acronym PEMDAS or BODMAS) tells us to handle exponents before addition. So, first, we need to calculate $9^2$, which means 9 multiplied by itself (9 * 9). Then, after calculating the exponent, we'll add 19 to the result. This approach simplifies the problem, making it easier to solve. Let's start with $9^2$. This is equal to 81. Now, we add 19 to 81. Therefore, $81 + 19 = 100$. So, the answer to the first expression is 100. It's that simple! Each step is a building block; once you understand one, the next step becomes easier. Breaking down the problem into smaller, manageable parts is key. It's like building with LEGOs; each piece adds to the structure. This is a fundamental concept in algebra, so understanding it will set you up for success with more complicated problems. Understanding exponents and addition in this context is essential. Exponents are a concise way of expressing repeated multiplication, and understanding this makes simplifying expressions a breeze. By breaking down the problem this way, you make it less overwhelming and much more manageable. The key is to take your time and make sure you do each step correctly.

Step-by-Step Solution

1. Calculate the exponent: $9^2 = 81$
2. Add the numbers: $81 + 19 = 100$

Answer

The value of the expression $9^2 + 19$ is 100.

б) $17^2 - 209$

Alright, let's tackle the next expression: $17^2 - 209$. This one involves subtraction, but it follows the same principles as the previous one. We start by dealing with the exponent. Again, we apply the order of operations. First, calculate $17^2$, which is 17 multiplied by itself (17 * 17). Then, we subtract 209 from that result. Using the order of operations simplifies the equation and reduces errors. This is crucial for maintaining accuracy in your calculations. Taking it slow and keeping track of each step helps prevent mistakes. Remember, algebra is like a puzzle: each piece fits together to create the final picture. Let's calculate $17^2$. We have $17 * 17 = 289$. Now we subtract 209 from 289. So, $289 - 209 = 80$. Great job! The answer to this expression is 80. Every step here is a move in the game, bringing you closer to the final solution. The more you practice, the easier it will become. Mastery comes from consistent effort and a clear understanding of each step. The order of operations ensures that you arrive at the correct answer. This ensures consistency and accuracy in solving complex equations. Mastering these fundamental operations lays the groundwork for tackling more complex algebraic equations, boosting your confidence along the way.

Step-by-Step Solution

1. Calculate the exponent: $17^2 = 289$
2. Subtract the numbers: $289 - 209 = 80$

Answer

The value of the expression $17^2 - 209$ is 80.

в) $6^3 : 3$

Okay, let's move on to the next one: $6^3 : 3$. This expression involves an exponent and division. Remember, the order of operations tells us to handle exponents before division. This structure simplifies your approach, ensuring each calculation is accurate. First, we need to calculate $6^3$, which means 6 multiplied by itself three times (6 * 6 * 6). After calculating the exponent, we'll divide the result by 3. This systematic approach is key to accuracy. Let's start with $6^3$. That is, 6 * 6 * 6 = 216. Next, we divide 216 by 3. So, $216 : 3 = 72$. Voila! The answer is 72. Breaking down the problem step-by-step removes the complexity and prevents mistakes. Mastering the fundamental operations of algebra like exponents and division is essential. Understanding the order of operations ensures consistent and accurate solutions. With consistent practice, these steps become second nature. Each problem helps solidify your understanding and builds confidence. Using this step-by-step approach not only ensures accuracy but also makes the process easier to follow and comprehend.

Step-by-Step Solution

1. Calculate the exponent: $6^3 = 216$
2. Divide the numbers: $216 : 3 = 72$

Answer

The value of the expression $6^3 : 3$ is 72.

г) $2^3 imes 3^2$

Let's get cracking with the next one: $2^3 imes 3^2$. This one brings together exponents and multiplication. Remember, we still follow the order of operations. We need to calculate each exponent separately before we do the multiplication. This systematic approach reduces the chance of errors. First, let's work on $2^3$, which is 2 * 2 * 2 = 8. Next, calculate $3^2$, which is 3 * 3 = 9. Finally, we multiply the two results. So, we have 8 * 9 = 72. Great job! The answer to this expression is 72. Solving these problems is like building a house. Each calculation is a brick that fits perfectly to give a solid structure. Understanding the order of operations is important. This ensures we perform calculations in the right sequence. The more you work through these problems, the more confident you'll feel in your algebraic skills. Always double-check your work to ensure accuracy. The ability to break down the equation into smaller and simpler steps is very important in algebra. Keep practicing, and you'll find that these expressions become more manageable and even fun to solve!

Step-by-Step Solution

1. Calculate the exponents: $2^3 = 8$ and $3^2 = 9$
2. Multiply the numbers: $8 imes 9 = 72$

Answer

The value of the expression $2^3 imes 3^2$ is 72.

д) $(15 - 7)^2 : 2^3$

Alright, let's move on to this one: $(15 - 7)^2 : 2^3$. This expression has parentheses, an exponent, and division. Remember, when you have parentheses, always handle what's inside the parentheses first. It is the first step. Then, apply the exponents, and finally, do the division. Following these rules systematically ensures accuracy. Inside the parentheses, we have $15 - 7$, which is 8. Now we have $8^2 : 2^3$. Next, calculate $8^2$, which is 8 * 8 = 64. Then, calculate $2^3$, which is 2 * 2 * 2 = 8. Finally, divide 64 by 8. So, $64 : 8 = 8$. You're doing great! The answer is 8. Breaking the problem down ensures you don't miss a step. The order of operations, including handling parentheses first, is vital. Remember, practice makes perfect. The more you work through these problems, the more comfortable you will become. Keep up the good work; you're doing fantastic! Each step builds upon the previous one, and with practice, you will master these skills. Confidence grows with each problem solved, and you're well on your way to becoming an algebra pro. This step-by-step approach simplifies the complex parts of the problems and makes it easy to understand.

Step-by-Step Solution

1. Solve the parentheses: $(15 - 7) = 8$
2. Calculate the exponents: $8^2 = 64$ and $2^3 = 8$
3. Divide the numbers: $64 : 8 = 8$

Answer

The value of the expression $(15 - 7)^2 : 2^3$ is 8.

е) $(17 - 16)^8 + 2^5$

Okay, let's solve the next expression: $(17 - 16)^8 + 2^5$. This one includes parentheses, exponents, and addition. Always start with the parentheses. This ensures that you're following the correct steps. Then, deal with the exponents, and finally, do the addition. Following a consistent approach reduces the likelihood of errors. Inside the parentheses, we have $17 - 16$, which equals 1. Now, we have $1^8 + 2^5$. The next step is to calculate $1^8$, which equals 1. And also, calculate $2^5$, which is 2 * 2 * 2 * 2 * 2 = 32. Finally, add the two results. So, $1 + 32 = 33$. Well done! The answer is 33. The key is to take it one step at a time, staying focused on each operation. Every step brings you closer to the solution. The consistent practice will build your confidence and sharpen your skills. With each solved problem, you deepen your understanding. This step-by-step approach breaks down the complex problem into easy and understandable parts. You are doing a fantastic job. The more you practice, the easier it will become. Always remember, consistency is key! Keep up the effort, and you'll become a pro in no time.

Step-by-Step Solution

1. Solve the parentheses: $(17 - 16) = 1$
2. Calculate the exponents: $1^8 = 1$ and $2^5 = 32$
3. Add the numbers: $1 + 32 = 33$

Answer

The value of the expression $(17 - 16)^8 + 2^5$ is 33.

ж) $10^6 - 20^4$

Alright, let's tackle this next one: $10^6 - 20^4$. This expression has exponents and subtraction. Remember, always deal with the exponents first. That means we have to calculate $10^6$ and $20^4$ before we can do the subtraction. This ensures that we perform operations in the correct order. First, calculate $10^6$, which means 10 multiplied by itself six times. That's $10 * 10 * 10 * 10 * 10 * 10 = 1,000,000$. Next, calculate $20^4$, which is $20 * 20 * 20 * 20 = 160,000$. Finally, we subtract the two results: $1,000,000 - 160,000 = 840,000$. Great work! The answer to this expression is 840,000. Each step helps you to understand how it's done. Understanding the order of operations will make all expressions easier. Keep practicing; with each problem, you're improving your skills. Always double-check your work to ensure accuracy. The ability to break down complex expressions into simple steps is invaluable. Remember, consistent effort and a clear understanding are essential for success. You are doing a great job! Keep up the good work!

Step-by-Step Solution

1. Calculate the exponents: $10^6 = 1,000,000$ and $20^4 = 160,000$
2. Subtract the numbers: $1,000,000 - 160,000 = 840,000$

Answer

The value of the expression $10^6 - 20^4$ is 840,000.

з) $3^4 imes 10^4$

Okay, let's try the following expression: $3^4 imes 10^4$. This expression includes exponents and multiplication. We need to handle the exponents first, following the order of operations. Then, we can multiply the two results. This approach ensures accuracy. Start with $3^4$, which means 3 multiplied by itself four times. That's $3 * 3 * 3 * 3 = 81$. Next, compute $10^4$, which is $10 * 10 * 10 * 10 = 10,000$. Finally, multiply the two results. So, we have $81 * 10,000 = 810,000$. Awesome! The answer is 810,000. Keep in mind that solving expressions like these is like a puzzle. Every operation you perform brings you closer to the final solution. Practice and understanding the order of operations are important. The more you practice, the more comfortable you become. Always take your time and double-check your work to catch any potential errors. Keep up the good work, you're doing great! You are on your way to mastering these skills.

Step-by-Step Solution

1. Calculate the exponents: $3^4 = 81$ and $10^4 = 10,000$
2. Multiply the numbers: $81 imes 10,000 = 810,000$

Answer

The value of the expression $3^4 imes 10^4$ is 810,000.

и) $5^4 : 5^2$

Alright, last but not least, let's tackle this expression: $5^4 : 5^2$. This one involves exponents and division. Remember the order of operations: First, deal with the exponents, then perform the division. This ensures accuracy. We start by calculating $5^4$, which is $5 * 5 * 5 * 5 = 625$. Next, compute $5^2$, which is $5 * 5 = 25$. Then, we divide $625$ by $25$. So, $625 : 25 = 25$. Fantastic! The answer is 25. Each expression we have worked through provides a building block for future problems. The key is to take it one step at a time, ensuring you understand each operation. Keep practicing, and you'll find these expressions easier and easier to solve. Always remember the order of operations, and you'll be on the right track. You've done an amazing job working through these problems. Continue with this effort, and your skills will only improve. Your hard work and dedication will pay off! Well done!

Step-by-Step Solution

1. Calculate the exponents: $5^4 = 625$ and $5^2 = 25$
2. Divide the numbers: $625 : 25 = 25$

Answer

The value of the expression $5^4 : 5^2$ is 25.