Multiply And Simplify 7√5(√10 + 8) A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of multiplying and simplifying the expression $7\sqrt{5}(\sqrt{10} + 8)$. This problem combines the concepts of the distributive property and simplification of radicals, providing an excellent opportunity to enhance your understanding of algebraic manipulations involving square roots. We will break down each step meticulously, ensuring clarity and precision in our approach. By the end of this article, you will not only be able to solve this particular problem but also gain a deeper insight into handling similar algebraic expressions. This is important as these skills form a cornerstone of more advanced mathematics, and mastering them will undoubtedly benefit you in future mathematical endeavors. Whether you are a student looking to improve your algebra skills or simply someone who enjoys mathematical challenges, this guide is designed to equip you with the knowledge and confidence to tackle such problems effectively. So, let's embark on this mathematical journey and unlock the secrets of simplifying expressions involving radicals.

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by a sum or difference inside parentheses. This property is crucial for expanding expressions and simplifying them into a more manageable form. In essence, the distributive property states that for any numbers $a$, $b$, and $c$, the following equation holds true:

$$a(b + c) = ab + ac$$

This means that we multiply the term outside the parentheses ($a$) by each term inside the parentheses ($b$ and $c$) and then add the results. The distributive property is not limited to addition; it also applies to subtraction:

$$a(b - c) = ab - ac$$

In this case, we multiply $a$ by both $b$ and $c$, but we subtract the product of $a$ and $c$ from the product of $a$ and $b$. This property is extremely versatile and is used extensively in simplifying algebraic expressions, solving equations, and much more. For example, consider the expression $3(x + 2)$. To simplify this, we apply the distributive property:

$$3(x + 2) = 3 imes x + 3 imes 2 = 3x + 6$$

Here, we multiplied 3 by both $x$ and 2, resulting in the simplified expression $3x + 6$. The distributive property is particularly useful when dealing with expressions involving radicals, as it allows us to break down complex expressions into simpler terms that can be further simplified. In the context of our problem, $7\sqrt{5}(\sqrt{10} + 8)$, we will use the distributive property to multiply $7\sqrt{5}$ by both $\sqrt{10}$ and 8, which will set the stage for simplifying the expression.

To begin simplifying the expression $7\sqrt{5}(\sqrt{10} + 8)$, we will apply the distributive property. This involves multiplying the term outside the parentheses, which is $7\sqrt{5}$, by each term inside the parentheses, $\sqrt{10}$ and 8. Let’s break this down step by step:

First, we multiply $7\sqrt{5}$ by $\sqrt{10}$:

$$7\sqrt{5} imes \sqrt{10} = 7(\sqrt{5} imes \sqrt{10})$$

Next, we multiply $7\sqrt{5}$ by 8:

$$7\sqrt{5} imes 8 = 7 imes 8 imes \sqrt{5} = 56\sqrt{5}$$

Now, we combine these two results using the distributive property:

$$7\sqrt{5}(\sqrt{10} + 8) = 7\sqrt{5} imes \sqrt{10} + 7\sqrt{5} imes 8 = 7(\sqrt{5} imes \sqrt{10}) + 56\sqrt{5}$$

This step is crucial because it transforms a single expression into a sum of two terms, each of which can be simplified individually. The first term, $7(\sqrt{5} imes \sqrt{10})$, involves the product of two square roots, which we can further simplify using the property $\sqrt{a} imes \sqrt{b} = \sqrt{ab}$. The second term, $56\sqrt{5}$, is already in a simplified form, as the square root of 5 cannot be further reduced. The next step will focus on simplifying the term involving the product of square roots, which will bring us closer to the final simplified expression. By meticulously applying the distributive property, we have laid the groundwork for further simplification and ensured that each term is correctly addressed.

After applying the distributive property, our expression now looks like this:

$$7(\sqrt{5} imes \sqrt{10}) + 56\sqrt{5}$$

The next step involves simplifying the term $7(\sqrt{5} imes \sqrt{10})$. To do this, we use the property that the product of square roots is the square root of the product, which is $\sqrt{a} imes \sqrt{b} = \sqrt{ab}$. Applying this property, we get:

$$\sqrt{5} imes \sqrt{10} = \sqrt{5 imes 10} = \sqrt{50}$$

Now, we need to simplify $\sqrt{50}$. We look for perfect square factors of 50. The largest perfect square factor of 50 is 25, since $50 = 25 imes 2$. Thus, we can rewrite $\sqrt{50}$ as:

$$\sqrt{50} = \sqrt{25 imes 2}$$

Using the property $\sqrt{ab} = \sqrt{a} imes \sqrt{b}$, we can further simplify this:

$$\sqrt{25 imes 2} = \sqrt{25} imes \sqrt{2} = 5\sqrt{2}$$

So, $\sqrt{50}$ simplifies to $5\sqrt{2}$. Now, we substitute this back into our original term:

$$7(\sqrt{5} imes \sqrt{10}) = 7(5\sqrt{2}) = 35\sqrt{2}$$

Now our expression looks like this:

$$35\sqrt{2} + 56\sqrt{5}$$

At this point, we have simplified the radicals as much as possible. The term $35\sqrt{2}$ has $\sqrt{2}$, which cannot be simplified further, and the term $56\sqrt{5}$ has $\sqrt{5}$, which also cannot be simplified further. The next step is to check if we can combine these terms, which depends on whether they have like radicals.

After simplifying the radicals, we have the expression:

$$35\sqrt{2} + 56\sqrt{5}$$

To determine if we can combine these terms, we need to check if they are like terms. Like terms in the context of radicals are terms that have the same radical part. In other words, they must have the same expression under the square root symbol.

In our expression, we have two terms: $35\sqrt{2}$ and $56\sqrt{5}$. The radical part of the first term is $\sqrt{2}$, and the radical part of the second term is $\sqrt{5}$. Since $\sqrt{2}$ and $\sqrt{5}$ are different, these terms are not like terms.

Like terms can be combined by adding or subtracting their coefficients. For example, $3\sqrt{7} + 5\sqrt{7}$ are like terms because they both have $\sqrt{7}$ as the radical part. We can combine them as follows:

$$3\sqrt{7} + 5\sqrt{7} = (3 + 5)\sqrt{7} = 8\sqrt{7}$$

However, since $35\sqrt{2}$ and $56\sqrt{5}$ are not like terms, we cannot combine them. They remain separate terms in our simplified expression. This means that we have reached the final stage of simplification for this expression. There are no further operations we can perform to reduce the expression to a simpler form.

Having meticulously applied the distributive property, simplified the radicals, and checked for like terms, we have arrived at the final simplified expression. Our original expression was:

$$7\sqrt{5}(\sqrt{10} + 8)$$

After applying the distributive property, we had:

$$7\sqrt{5} imes \sqrt{10} + 7\sqrt{5} imes 8$$

Which simplified to:

$$7(\sqrt{5} imes \sqrt{10}) + 56\sqrt{5}$$

We then simplified the radicals:

$$7\sqrt{50} + 56\sqrt{5}$$

$$\sqrt{50} = \sqrt{25 imes 2} = 5\sqrt{2}$$

So, the expression became:

$$7(5\sqrt{2}) + 56\sqrt{5}$$

$$35\sqrt{2} + 56\sqrt{5}$$

Finally, we checked for like terms and found that $35\sqrt{2}$ and $56\sqrt{5}$ cannot be combined because they have different radicals. Therefore, the final simplified expression is:

$$35\sqrt{2} + 56\sqrt{5}$$

This is the most simplified form of the original expression. We have successfully broken down the problem into manageable steps, applied the necessary properties of algebra and radicals, and arrived at the final answer. This result showcases the importance of understanding fundamental algebraic principles and the step-by-step approach to problem-solving in mathematics. The ability to simplify complex expressions like this is a valuable skill that can be applied in various mathematical contexts.

In this detailed walkthrough, we have successfully multiplied and simplified the expression $7\sqrt{5}(\sqrt{10} + 8)$. We began by understanding the distributive property, which allowed us to expand the expression into simpler terms. We then applied this property to multiply $7\sqrt{5}$ by both $\sqrt{10}$ and 8, resulting in $7(\sqrt{5} imes \sqrt{10}) + 56\sqrt{5}$.

Next, we focused on simplifying the radicals. We used the property $\sqrt{a} imes \sqrt{b} = \sqrt{ab}$ to combine the square roots and then simplified $\sqrt{50}$ to $5\sqrt{2}$. This transformation led to the expression $35\sqrt{2} + 56\sqrt{5}$.

Finally, we examined whether we could combine the terms. We determined that $35\sqrt{2}$ and $56\sqrt{5}$ are not like terms because they have different radicals, $\sqrt{2}$ and $\sqrt{5}$, respectively. Therefore, no further simplification was possible.

Our final simplified expression is:

$$35\sqrt{2} + 56\sqrt{5}$$

This exercise highlights the significance of mastering fundamental algebraic properties and techniques. The ability to simplify expressions involving radicals is a crucial skill in algebra and beyond. By understanding and applying these principles, you can confidently tackle similar mathematical challenges. This step-by-step approach not only helps in solving problems accurately but also enhances your overall mathematical reasoning and problem-solving abilities. We hope this guide has provided you with a clear and comprehensive understanding of how to multiply and simplify such expressions, equipping you with the tools to excel in your mathematical endeavors.