Simplifying Radicals: $10√10 - 7√10$ Explained

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In this article, we'll break down how to simplify the expression $10\sqrt{10} - 7\sqrt{10}$. This involves understanding how to work with radicals and perform basic arithmetic operations with them. Let's dive in!

Understanding the Basics of Radicals

Before we tackle the main problem, let's quickly recap what radicals are. A radical is a mathematical expression that involves a root, such as a square root, cube root, or higher root. The most common type is the square root, denoted by the symbol $\sqrt{}$. For example, $\sqrt{9} = 3$ because 3 multiplied by itself equals 9.

When dealing with radicals, it's important to remember that you can only combine like terms. In the context of radicals, "like terms" mean terms that have the same radical part. For instance, $5\sqrt{2}$ and $3\sqrt{2}$ are like terms because they both have $\sqrt{2}$ as the radical part. However, $5\sqrt{2}$ and $3\sqrt{3}$ are not like terms because they have different radical parts.

Breaking Down the Components

To simplify the expression $10\sqrt{10} - 7\sqrt{10}$, we need to recognize the components involved. Here, we have two terms: $10\sqrt{10}$ and $7\sqrt{10}$.

• Term 1: $10\sqrt{10}$ consists of a coefficient (10) and a radical part ($\sqrt{10}$). This means we have 10 times the square root of 10.
• Term 2: $7\sqrt{10}$ consists of a coefficient (7) and a radical part ($\sqrt{10}$). This means we have 7 times the square root of 10.

Since both terms have the same radical part, $\sqrt{10}$, they are like terms, and we can combine them.

Step-by-Step Simplification

Now that we understand the basics, let's simplify the expression step by step.

Step 1: Identify Like Terms

As we've already established, the terms $10\sqrt{10}$ and $7\sqrt{10}$ are like terms because they both contain the radical $\sqrt{10}$.

Step 2: Combine Like Terms

To combine like terms, we simply add or subtract their coefficients while keeping the radical part the same. In this case, we subtract the coefficients:

$10\sqrt{10} - 7\sqrt{10} = (10 - 7)\sqrt{10}$

Step 3: Perform the Arithmetic

Now, perform the subtraction:

$10 - 7 = 3$

So, the expression becomes:

$3\sqrt{10}$

Final Result

Therefore, the simplified form of $10\sqrt{10} - 7\sqrt{10}$ is $3\sqrt{10}$.

Practical Examples

Let's go through a few more examples to solidify your understanding.

Example 1: $5\sqrt{3} + 2\sqrt{3}$

Here, we have two terms: $5\sqrt{3}$ and $2\sqrt{3}$. Both terms have the same radical part, $\sqrt{3}$, so they are like terms. To combine them, we add their coefficients:

$5\sqrt{3} + 2\sqrt{3} = (5 + 2)\sqrt{3} = 7\sqrt{3}$

Example 2: $8\sqrt{5} - 3\sqrt{5}$

In this case, we have $8\sqrt{5}$ and $3\sqrt{5}$. Again, both terms have the same radical part, $\sqrt{5}$, so they are like terms. We subtract their coefficients:

$8\sqrt{5} - 3\sqrt{5} = (8 - 3)\sqrt{5} = 5\sqrt{5}$

Example 3: $4\sqrt{7} + 6\sqrt{7} - \sqrt{7}$

Here, we have three terms: $4\sqrt{7}$, $6\sqrt{7}$, and $-\sqrt{7}$. All terms have the same radical part, $\sqrt{7}$, so they are like terms. Note that $-\sqrt{7}$ can be thought of as $-1\sqrt{7}$. We combine the coefficients:

$4\sqrt{7} + 6\sqrt{7} - \sqrt{7} = (4 + 6 - 1)\sqrt{7} = 9\sqrt{7}$

Common Mistakes to Avoid

When simplifying radical expressions, it's easy to make a few common mistakes. Here are some to watch out for:

1. Combining Unlike Terms: Only combine terms that have the same radical part. For example, you cannot combine $5\sqrt{2}$ and $3\sqrt{3}$.
2. Incorrect Arithmetic: Double-check your addition and subtraction of coefficients to avoid errors.
3. Forgetting the Radical: When combining like terms, make sure to keep the radical part of the term. For example, $5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2}$, not just 8.
4. Not Simplifying Radicals: Before combining terms, ensure that each radical is simplified. For example, $\sqrt{8}$ can be simplified to $2\sqrt{2}$ before combining it with other terms.

Advanced Tips and Tricks

Simplifying Radicals First

Sometimes, you may need to simplify the radicals before you can combine like terms. For example, consider the expression $\sqrt{8} + \sqrt{18}$. At first glance, it may seem like these terms cannot be combined because they have different radical parts. However, we can simplify each radical:

$\sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$

$\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}$

Now, we can rewrite the expression as:

$2\sqrt{2} + 3\sqrt{2}$

These are like terms, so we can combine them:

$2\sqrt{2} + 3\sqrt{2} = (2 + 3)\sqrt{2} = 5\sqrt{2}$

Rationalizing the Denominator

Another useful technique is rationalizing the denominator. This involves removing any radicals from the denominator of a fraction. For example, consider the fraction $\frac{1}{\sqrt{2}}$. To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{2}$:

$\frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

This technique is useful when you need to simplify expressions involving fractions with radicals in the denominator.

Real-World Applications

Understanding how to simplify radical expressions is not just a theoretical exercise. It has practical applications in various fields, including:

1. Physics: Radicals often appear in physics equations, such as those involving energy, motion, and waves. Simplifying these expressions can make calculations easier and more accurate.
2. Engineering: Engineers use radicals in various calculations, such as determining the strength of materials, analyzing electrical circuits, and designing structures.
3. Computer Graphics: Radicals are used in computer graphics to calculate distances, angles, and other geometric properties. Simplifying these expressions can improve the performance of graphics applications.
4. Finance: Financial analysts use radicals in various calculations, such as determining the rate of return on investments and analyzing risk.

Conclusion

Simplifying expressions like $10\sqrt{10} - 7\sqrt{10}$ involves understanding the basic principles of radicals and how to combine like terms. By identifying the components, combining the coefficients, and keeping the radical part the same, you can easily simplify these expressions. Remember to avoid common mistakes and to simplify radicals before combining terms. With practice, you'll become proficient in simplifying radical expressions and applying them in various real-world scenarios.

Further enhance your understanding of radicals and mathematical operations by exploring resources on trusted educational websites such as Khan Academy's Algebra I section.