Simplifying Algebraic Expressions And Finding Values

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Hey guys! Today, we're diving deep into the fascinating world of algebra, where we'll be simplifying expressions and plugging in values to find solutions. This might sound intimidating, but trust me, it's like solving a puzzle – super rewarding when you crack it! We’ll break down each problem step-by-step, so you can follow along easily. Let's get started and make algebra less of a mystery and more of a fun challenge.

1) Simplifying (20a⁷ + 7a³) - (57 + 20a⁷) when a = 2

Let's start with our first expression: (20a⁷ + 7a³) - (57 + 20a⁷). Our goal here is to simplify this expression and then find its value when a equals 2. First things first, we need to get rid of those parentheses. Remember, when we have a minus sign in front of a parenthesis, it's like we're distributing a -1 to everything inside.

So, let’s rewrite the expression:

20a⁷ + 7a³ - 57 - 20a⁷

Now, we need to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have 20a⁷ and -20a⁷. Notice anything? They cancel each other out! This is because 20a⁷ - 20a⁷ = 0. So, we can simply eliminate them from the equation. What we're left with now is:

7a³ - 57

Awesome! We've simplified the expression. The next step is to substitute a with 2. This means everywhere we see a, we replace it with 2. Our expression becomes:

7(2)³ - 57

Now, we need to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we have an exponent, so we tackle that first. 2³ means 2 * 2 * 2, which equals 8. So, we can replace 2³ with 8:

7 * 8 - 57

Next up is multiplication: 7 * 8 = 56. Our expression now looks like:

56 - 57

Finally, we have subtraction. 56 - 57 equals -1. So, the value of the expression (20a⁷ + 7a³) - (57 + 20a⁷) when a = 2 is -1.

See? Not so scary, right? We took it one step at a time, simplified, substituted, and solved. Remember, breaking down a problem into smaller steps makes it much more manageable. Now, let’s move on to the next one.

2) Simplifying (17.3x⁵ + 62) + (3x² - 17.3x⁵) when x = -5

Okay, let's tackle the second expression: (17.3x⁵ + 62) + (3x² - 17.3x⁵). This time, we're dealing with decimals and a negative value for x, but don’t sweat it! We’ll use the same principles we learned in the first problem.

First, let’s get rid of the parentheses. Since we have a plus sign between the parentheses, it’s pretty straightforward – we can just drop them. The expression becomes:

  1. 3x⁵ + 62 + 3x² - 17.3x⁵

Now, let's look for those like terms again. We have 17.3x⁵ and -17.3x⁵. Just like before, these terms cancel each other out because 17.3x⁵ - 17.3x⁵ = 0. This leaves us with:

62 + 3x²

Notice that we can also write this as:

3x² + 62

This is often preferred because it follows the standard form of writing polynomials (with the highest power of the variable first). Now that we've simplified the expression, it's time to substitute x with -5. Remember, we need to be careful with negative numbers and exponents. Our expression becomes:

3(-5)² + 62

Following the order of operations, we need to deal with the exponent first. (-5)² means -5 * -5, which equals 25. A negative number multiplied by a negative number gives a positive result. So, we replace (-5)² with 25:

3 * 25 + 62

Next, we do the multiplication: 3 * 25 = 75. Our expression now looks like:

75 + 62

Finally, we add 75 and 62, which gives us 137. So, the value of the expression (17.3x⁵ + 62) + (3x² - 17.3x⁵) when x = -5 is 137.

We successfully navigated through decimals, negative numbers, and exponents! Remember, the key is to take your time, pay close attention to the signs, and follow the order of operations. We're halfway there – let’s keep the momentum going!

3) Simplifying (83.61 + 9.1b) - (2.763 + 8.75b) when b = 4

Alright, let's jump into the third expression: (83.61 + 9.1b) - (2.763 + 8.75b). We're continuing to build our skills here, and this one has decimals and a subtraction between the parentheses, so let's break it down step-by-step.

As we did before, our first task is to remove the parentheses. Remember, the minus sign in front of the second set of parentheses means we need to distribute a -1 to each term inside. So, the expression becomes:

  1. 61 + 9.1b - 2.763 - 8.75b

Now, let's identify and combine those like terms. We have two sets of like terms here: the constant terms (83.61 and -2.763) and the terms with the variable b (9.1b and -8.75b). Let's group them together:

(83.61 - 2.763) + (9.1b - 8.75b)

Now, let's perform the subtractions. First, 83.61 - 2.763 = 80.847. Then, 9.1b - 8.75b = 0.35b. So, our simplified expression is:

  1. 847 + 0.35b

Great job! We’ve simplified the expression. Now, let's substitute b with 4. Our expression becomes:

  1. 847 + 0.35 * 4

Following the order of operations, we need to do the multiplication first: 0.35 * 4 = 1.4. So, the expression becomes:

  1. 847 + 1.4

Finally, we add 80.847 and 1.4, which gives us 82.247. So, the value of the expression (83.61 + 9.1b) - (2.763 + 8.75b) when b = 4 is 82.247.

We're doing fantastic! We handled the subtraction between parentheses and worked with decimals like pros. Just one more problem to go – let’s keep up the great work!

4) Simplifying (14y - 11.3) + (6y² + 11.3y) when y = -3/7

Okay, last but not least, we have the expression: (14y - 11.3) + (6y² + 11.3y), where y equals -3/7. This one might look a little trickier with the fraction, but we've got the skills to handle it! The same rules apply: simplify the expression first and then substitute the value of y.

First, let’s get rid of the parentheses. Since we have a plus sign between the parentheses, we can simply drop them. The expression becomes:

14y - 11.3 + 6y² + 11.3y

Now, let’s identify and combine those like terms. We have two terms with y: 14y and 11.3y. Let's combine them and also rearrange the terms so that the term with the highest power of y comes first. This gives us:

6y² + 14y + 11.3y - 11.3

Combining 14y and 11.3y, we get 25.3y. So, our simplified expression is:

6y² + 25.3y - 11.3

Now comes the fun part – substituting y with -3/7. This means we replace every y in the expression with -3/7. Our expression becomes:

6(-3/7)² + 25.3(-3/7) - 11.3

Following the order of operations, we start with the exponent. (-3/7)² means (-3/7) * (-3/7), which equals 9/49 (a negative times a negative is a positive). So, we replace (-3/7)² with 9/49:

6(9/49) + 25.3(-3/7) - 11.3

Next, we perform the multiplications. 6 * (9/49) = 54/49, and 25.3 * (-3/7) = -75.9/7. Our expression now looks like:

(54/49) - (75.9/7) - 11.3

This is where things get a little trickier with the fractions and decimals. To make things easier, let's convert the fractions to decimals (approximately). 54/49 is about 1.102, and -75.9/7 is about -10.843. So, our expression becomes:

  1. 102 - 10.843 - 11.3

Now, we perform the subtractions from left to right. 1.102 - 10.843 = -9.741, and then -9.741 - 11.3 = -21.041. So, the value of the expression (14y - 11.3) + (6y² + 11.3y) when y = -3/7 is approximately -21.041.

Conclusion

Woohoo! We made it through all four expressions! Give yourselves a pat on the back. We’ve tackled parentheses, combined like terms, handled decimals and fractions, worked with negative numbers, and followed the order of operations. Remember, practice makes perfect, so the more you work with these types of problems, the more comfortable and confident you’ll become.

Algebra might seem like a tough nut to crack at first, but by breaking it down into smaller, manageable steps, it becomes much less daunting. Keep practicing, keep asking questions, and you’ll be simplifying expressions like a pro in no time! You've got this!