Simplify.$\sqrt{5}(\sqrt{125}+\sqrt{10}$\]

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems more efficiently. When dealing with square roots, it's essential to understand the properties of radicals and how to manipulate them. In this article, we will simplify the given expression $\sqrt{5}(\sqrt{125}+\sqrt{10})$ using various techniques and properties of radicals.

Understanding the Expression

The given expression is a product of two square roots, $\sqrt{5}$ and $(\sqrt{125}+\sqrt{10})$. To simplify this expression, we need to understand the properties of radicals and how to manipulate them. The expression can be broken down into two parts: $\sqrt{5}$ and $(\sqrt{125}+\sqrt{10})$. We will simplify each part separately and then combine them.

Simplifying $\sqrt{125}$

To simplify $\sqrt{125}$, we need to find the largest perfect square that divides 125. We know that $125 = 25 \times 5$, and since $25$ is a perfect square, we can write $\sqrt{125}$ as $\sqrt{25 \times 5}$. Using the property of radicals, we can simplify this as $\sqrt{25} \times \sqrt{5} = 5\sqrt{5}$.

Simplifying $\sqrt{10}$

To simplify $\sqrt{10}$, we need to find the largest perfect square that divides 10. We know that $10 = 1 \times 10$, and since $1$ is a perfect square, we can write $\sqrt{10}$ as $\sqrt{1 \times 10}$. Using the property of radicals, we can simplify this as $\sqrt{1} \times \sqrt{10} = \sqrt{10}$.

Combining the Simplified Expressions

Now that we have simplified $\sqrt{125}$ and $\sqrt{10}$, we can combine them with $\sqrt{5}$ to get the final simplified expression. We have:

$\sqrt{5}(\sqrt{125}+\sqrt{10}) = \sqrt{5}(5\sqrt{5}+\sqrt{10})$

Distributing $\sqrt{5}$

To simplify the expression further, we need to distribute $\sqrt{5}$ to both terms inside the parentheses. We have:

$\sqrt{5}(5\sqrt{5}+\sqrt{10}) = 5\sqrt{5} \times \sqrt{5} + \sqrt{5} \times \sqrt{10}$

Simplifying the Expression

Using the property of radicals, we can simplify the expression as:

$5\sqrt{5} \times \sqrt{5} + \sqrt{5} \times \sqrt{10} = 25 + \sqrt{50}$

Simplifying $\sqrt{50}$

To simplify $\sqrt{50}$, we need to find the largest perfect square that divides 50. We know that $50 = 25 \times 2$, and since $25$ is a perfect square, we can write $\sqrt{50}$ as $\sqrt{25 \times 2}$. Using the property of radicals, we can simplify this as $\sqrt{25} \times \sqrt{2} = 5\sqrt{2}$.

Final Simplified Expression

Now that we have simplified $\sqrt{50}$, we can combine it with the other terms to get the final simplified expression. We have:

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

Conclusion

In this article, we simplified the given expression $\sqrt{5}(\sqrt{125}+\sqrt{10})$ using various techniques and properties of radicals. We broke down the expression into two parts, simplified each part separately, and then combined them to get the final simplified expression. The final simplified expression is $25 + 5\sqrt{2}$.

Properties of Radicals

Radicals are a fundamental concept in mathematics, and understanding their properties is essential for simplifying expressions. Some of the key properties of radicals include:

• Product of Radicals: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
• Quotient of Radicals: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
• Power of a Radical: $(\sqrt{a})^n = a^{\frac{n}{2}}$

Examples of Simplifying Expressions

Simplifying expressions is a crucial skill in mathematics, and there are many examples of how to simplify expressions using radicals. Some examples include:

• Simplifying $\sqrt{16} + \sqrt{9}$: $\sqrt{16} + \sqrt{9} = 4 + 3 = 7$
• Simplifying $\sqrt{36} - \sqrt{25}$: $\sqrt{36} - \sqrt{25} = 6 - 5 = 1$
• Simplifying $\sqrt{49} \times \sqrt{9}$: $\sqrt{49} \times \sqrt{9} = 7 \times 3 = 21$

Applications of Simplifying Expressions

Simplifying expressions has many applications in mathematics and other fields. Some examples include:

• Algebra: Simplifying expressions is a crucial skill in algebra, where we use radicals to solve equations and inequalities.
• Geometry: Simplifying expressions is used in geometry to find the area and perimeter of shapes.
• Physics: Simplifying expressions is used in physics to solve problems involving motion and energy.

Final Thoughts

Simplifying expressions is a fundamental skill in mathematics, and understanding the properties of radicals is essential for simplifying expressions. In this article, we simplified the given expression $\sqrt{5}(\sqrt{125}+\sqrt{10})$ using various techniques and properties of radicals. We broke down the expression into two parts, simplified each part separately, and then combined them to get the final simplified expression. The final simplified expression is $25 + 5\sqrt{2}$.

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Introduction

In our previous article, we simplified the given expression $\sqrt{5}(\sqrt{125}+\sqrt{10})$ using various techniques and properties of radicals. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions with radicals.

Q: What is the difference between a radical and a rational number?

A: A radical is an expression that involves a square root, such as $\sqrt{a}$, while a rational number is a number that can be expressed as the ratio of two integers, such as $\frac{a}{b}$.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you need to find the largest perfect square that divides the number inside the radical. You can then use the property of radicals to simplify the expression.

Q: What is the property of radicals that states $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$?

A: This property is known as the product of radicals. It states that when you multiply two radicals, you can combine them into a single radical by multiplying the numbers inside the radicals.

Q: How do I simplify an expression with a quotient of radicals?

A: To simplify an expression with a quotient of radicals, you need to find the largest perfect square that divides the number inside the radical. You can then use the property of radicals to simplify the expression.

Q: What is the property of radicals that states $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$?

A: This property is known as the quotient of radicals. It states that when you divide two radicals, you can combine them into a single radical by dividing the numbers inside the radicals.

Q: How do I simplify an expression with a power of a radical?

A: To simplify an expression with a power of a radical, you need to use the property of radicals that states $(\sqrt{a})^n = a^{\frac{n}{2}}$.

Q: What is the property of radicals that states $(\sqrt{a})^n = a^{\frac{n}{2}}$?

A: This property is known as the power of a radical. It states that when you raise a radical to a power, you can simplify the expression by raising the number inside the radical to that power.

Q: Can I simplify an expression with a radical that has a negative number inside?

A: Yes, you can simplify an expression with a radical that has a negative number inside. However, you need to be careful when simplifying the expression, as the negative sign may affect the final result.

Q: How do I simplify an expression with a radical that has a decimal number inside?

A: To simplify an expression with a radical that has a decimal number inside, you need to find the largest perfect square that divides the decimal number. You can then use the property of radicals to simplify the expression.

Q: Can I simplify an expression with a radical that has a variable inside?

A: Yes, you can simplify an expression with a radical that has a variable inside. However, you need to be careful when simplifying the expression, as the variable may affect the final result.

Q: How do I simplify an expression with a radical that has a fraction inside?

A: To simplify an expression with a radical that has a fraction inside, you need to find the largest perfect square that divides the fraction. You can then use the property of radicals to simplify the expression.

Q: Can I simplify an expression with a radical that has a negative fraction inside?

A: Yes, you can simplify an expression with a radical that has a negative fraction inside. However, you need to be careful when simplifying the expression, as the negative sign may affect the final result.

Q: How do I simplify an expression with a radical that has a complex number inside?

A: To simplify an expression with a radical that has a complex number inside, you need to use the property of radicals that states $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$. You can then simplify the expression by combining the complex numbers inside the radicals.

Q: Can I simplify an expression with a radical that has a matrix inside?

A: Yes, you can simplify an expression with a radical that has a matrix inside. However, you need to be careful when simplifying the expression, as the matrix may affect the final result.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions with radicals. We covered topics such as the difference between a radical and a rational number, the property of radicals that states $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$, and how to simplify expressions with radicals that have negative numbers, decimal numbers, variables, fractions, and complex numbers inside. We also covered how to simplify expressions with radicals that have matrices inside.