Distributive Property: Simplify 7b^5(9-10b^2) Easily

by Andrew McMorgan 53 views

Hey math enthusiasts! Ever find yourself staring at an expression like 7b5(9−10b2)7b^5(9-10b^2) and wondering how to simplify it? Don't sweat it! We're going to break it down step-by-step using the distributive property. Trust me, it's way easier than it looks. So, grab your pencils, and let's dive in!

Understanding the Distributive Property

Before we tackle our specific problem, let's quickly recap what the distributive property actually is. At its core, the distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms inside parentheses. Think of it as a way to "distribute" the multiplication across the addition or subtraction within the parentheses.

In simpler terms, the distributive property states that for any numbers a, b, and c:

a(b + c) = ab + ac

a(b - c) = ab - ac

This means you multiply the term outside the parentheses (a) by each term inside the parentheses (b and c) separately and then add or subtract the results. This property is crucial for simplifying algebraic expressions and solving equations, and it's one of those mathematical tools you'll use again and again. The key is to ensure you multiply the term outside the parenthesis with every term inside. One way to visually understand this is to draw "arrows" from the term outside the parentheses to each term inside, reminding you to perform each multiplication. Remembering the sign (positive or negative) of each term is also super important for getting the correct answer.

For example, if we have 3(x + 2), we distribute the 3 to both the x and the 2:

3 * x = 3x

3 * 2 = 6

So, 3(x + 2) simplifies to 3x + 6. See? Not too scary, right? Now, let's apply this knowledge to our main problem.

Applying the Distributive Property to 7b5(9−10b2)7b^5(9-10b^2)

Okay, let's get our hands dirty with the expression 7b5(9−10b2)7b^5(9-10b^2). Our mission is to simplify this using the distributive property, and it's totally doable. First, identify the term outside the parentheses, which is 7b57b^5, and the terms inside the parentheses, which are 9 and −10b2-10b^2. Remember that minus sign – it's super important! We need to distribute 7b57b^5 to both 9 and −10b2-10b^2. This means we'll perform two multiplications:

Multiply 7b57b^5 by 9

Multiply 7b57b^5 by −10b2-10b^2

Let’s tackle the first multiplication: 7b5∗97b^5 * 9. When multiplying terms with coefficients and variables, we multiply the coefficients (the numbers) and keep the variable part the same. So, 7 multiplied by 9 is 63. The variable part is just b5b^5, so the result is 63b563b^5. Now, let's move on to the second multiplication: 7b5∗−10b27b^5 * -10b^2. Again, we multiply the coefficients: 7 multiplied by -10 is -70. Now for the variables. We have b5b^5 multiplied by b2b^2. Remember the rule for multiplying exponents with the same base? We add the exponents! So, b5∗b2=b(5+2)=b7b^5 * b^2 = b^(5+2) = b^7. Combining the coefficient and the variable part, we get −70b7-70b^7.

So far, we've distributed 7b57b^5 to both terms inside the parentheses. Now we just need to combine the results. We have 63b563b^5 from the first multiplication and −70b7-70b^7 from the second multiplication. Putting it all together, the simplified expression is 63b5−70b763b^5 - 70b^7.

Step-by-Step Solution

Let's break it down into a clear, step-by-step solution so you can see exactly what we did:

  1. Identify the terms: We have 7b57b^5 outside the parentheses and (9−10b2)(9-10b^2) inside.
  2. Distribute: Multiply 7b57b^5 by each term inside the parentheses:
    • 7b5∗9=63b57b^5 * 9 = 63b^5
    • 7b5∗−10b2=−70b77b^5 * -10b^2 = -70b^7
  3. Combine the results: Write the results together: 63b5−70b763b^5 - 70b^7.

And that's it! We've successfully used the distributive property to simplify the expression. The final simplified form is 63b5−70b763b^5 - 70b^7. You might be tempted to try and combine these terms further, but remember, you can only combine terms that have the same variable and the same exponent. In this case, b5b^5 and b7b^7 are different, so we can't combine them.

Ordering the Terms (Optional)

While 63b5−70b763b^5 - 70b^7 is perfectly correct, it's often considered good mathematical practice to write the terms in descending order of their exponents. This means we would write the term with the highest exponent first. In our case, −70b7-70b^7 has a higher exponent (7) than 63b563b^5 (exponent 5). So, we can rewrite the expression as:

−70b7+63b5-70b^7 + 63b^5

This form is mathematically equivalent to 63b5−70b763b^5 - 70b^7, but it's just a slightly more polished way to present the answer. Think of it as the difference between wearing a t-shirt and wearing a nicely pressed shirt – both keep you covered, but one looks a bit more professional.

Common Mistakes to Avoid

Now that we've successfully simplified the expression, let's chat about some common pitfalls that students often encounter when using the distributive property. Being aware of these mistakes can help you avoid them in your own work!

  • Forgetting to distribute to all terms: This is probably the most common mistake. Remember, you need to multiply the term outside the parentheses by every term inside. If there are three terms inside, you need to perform three multiplications. Don't leave anyone out!
  • Incorrectly handling signs: Pay close attention to the signs (positive or negative) of the terms. A negative sign in front of a term inside the parentheses means you're multiplying by a negative number, which can change the sign of the resulting term. For example, as we saw, 7b5∗−10b27b^5 * -10b^2 results in a negative term, −70b7-70b^7.
  • Mistakes with exponents: When multiplying variables with exponents, remember to add the exponents, not multiply them. So, b5∗b2=b(5+2)=b7b^5 * b^2 = b^(5+2) = b^7, not b(5∗2)=b10b^(5*2) = b^10. It’s an easy mistake to make if you're rushing, so take your time and double-check!
  • Combining unlike terms: You can only combine terms that have the same variable and the same exponent. For instance, you can't combine 63b563b^5 and −70b7-70b^7 because the exponents are different. These are unlike terms, and they stay separate.
  • Forgetting the distributive property altogether: Sometimes, in the heat of the moment, students forget about the distributive property and try to do something completely different with the expression, like adding terms inside the parentheses first (which you can't do in this case because they are not like terms). Always remember the order of operations (PEMDAS/BODMAS) and the specific rules like the distributive property.

By keeping these common mistakes in mind, you'll be well-equipped to tackle distributive property problems with confidence and accuracy.

Practice Makes Perfect

The best way to master the distributive property is, you guessed it, practice! The more you work through different examples, the more comfortable and confident you'll become. Try working through similar problems with different coefficients and exponents. You can even create your own practice problems to challenge yourself.

For example, try simplifying these expressions:

5x3(2x−7)5x^3(2x - 7)

−3a2(4a3+6)-3a^2(4a^3 + 6)

8y4(3−9y2)8y^4(3 - 9y^2)

Work through them step-by-step, paying close attention to the signs, exponents, and the order of operations. Check your answers to make sure you're on the right track. If you get stuck, revisit the steps we discussed earlier, or ask a friend, teacher, or online resource for help. There are tons of fantastic resources available online, like Khan Academy and other math websites, that offer practice problems and video explanations. Don't hesitate to use them!

Real-World Applications

You might be wondering, "Okay, this is cool, but where am I ever going to use the distributive property in real life?" That's a valid question! While you might not be simplifying algebraic expressions every day in your daily routine, the distributive property is a foundational concept that underlies many mathematical and real-world applications. It's not just about the specific skill of expanding expressions; it's about building a solid algebraic foundation that will serve you well in more advanced math courses and in various problem-solving scenarios.

Here are a few examples of how the distributive property can pop up in everyday situations:

  • Calculating costs: Imagine you're buying 3 notebooks that cost $2 each and 3 pens that cost $1 each. You could calculate the total cost by finding the cost of the notebooks (3 * $2 = $6) and the cost of the pens (3 * $1 = $3) separately, and then adding them together ($6 + $3 = $9). Or, you could use the distributive property: 3($2 + $1) = 3 * $2 + 3 * $1 = $6 + $3 = $9. In this case, the distributive property provides a more efficient way to organize the calculation.
  • Home improvement projects: Let's say you're planning to tile a rectangular floor. If you know the dimensions of the floor and want to calculate the total area, you might break the floor into smaller sections. The distributive property can help you calculate the total area by distributing the dimensions. For instance, if the floor is (x + 2) feet long and 5 feet wide, the area is 5(x + 2) = 5x + 10 square feet.
  • Financial calculations: The distributive property is used in various financial calculations, such as calculating simple interest or compound interest. For example, if you invest P dollars at an annual interest rate of r for t years, the total amount you'll have (assuming simple interest) is P(1 + rt). Distributing the P gives you P + Prt, where P is your initial investment and Prt is the interest earned.

These are just a few examples, but the point is that the distributive property is a versatile tool that can be applied in many different contexts. By mastering this concept, you're not just learning a mathematical rule; you're developing a valuable problem-solving skill that will benefit you in various aspects of life.

Conclusion

So, there you have it! We've successfully simplified 7b5(9−10b2)7b^5(9-10b^2) using the distributive property. Remember, the key is to distribute the term outside the parentheses to every term inside, paying close attention to signs and exponents. Keep practicing, and you'll become a pro at using the distributive property in no time! You got this!