Simplify: 36 + 64 \sqrt{36+64} 36 + 64 ​ .

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Introduction

When dealing with mathematical expressions involving square roots, it's essential to simplify them to their most basic form. This not only makes the expression easier to understand but also helps in performing calculations and solving equations. In this article, we will simplify the expression $\sqrt{36+64}$ using basic algebraic operations and properties of square roots.

Understanding the Expression

The given expression is $\sqrt{36+64}$. To simplify this, we need to first evaluate the expression inside the square root. This involves adding the numbers 36 and 64.

Adding 36 and 64

To add 36 and 64, we simply perform the addition operation.

$\sqrt{36+64} = \sqrt{100}$

Simplifying the Square Root

Now that we have the expression inside the square root as 100, we can simplify it further. The square root of 100 is a number that, when multiplied by itself, gives 100.

Finding the Square Root of 100

To find the square root of 100, we can use the fact that the square root of a number is a value that, when multiplied by itself, gives the original number.

$\sqrt{100} = 10$

Conclusion

In this article, we simplified the expression $\sqrt{36+64}$ using basic algebraic operations and properties of square roots. We first evaluated the expression inside the square root by adding 36 and 64, and then simplified the square root of 100 to get the final answer.

Properties of Square Roots

Square roots have several properties that can be used to simplify expressions. Some of these properties include:

Property 1: Square Root of a Number is a Number

The square root of a number is a value that, when multiplied by itself, gives the original number.

Property 2: Square Root of a Product is the Product of the Square Roots

The square root of a product of two numbers is equal to the product of the square roots of the individual numbers.

Property 3: Square Root of a Sum is Not Equal to the Sum of the Square Roots

The square root of a sum of two numbers is not equal to the sum of the square roots of the individual numbers.

Examples of Simplifying Square Roots

Here are a few examples of simplifying square roots using the properties mentioned above.

Example 1: Simplifying $\sqrt{16+25}$

To simplify $\sqrt{16+25}$, we first evaluate the expression inside the square root by adding 16 and 25.

$\sqrt{16+25} = \sqrt{41}$

Since 41 is not a perfect square, we cannot simplify the square root further.

Example 2: Simplifying $\sqrt{9+16}$

To simplify $\sqrt{9+16}$, we first evaluate the expression inside the square root by adding 9 and 16.

$\sqrt{9+16} = \sqrt{25}$

Since 25 is a perfect square, we can simplify the square root further.

$\sqrt{25} = 5$

Real-World Applications of Simplifying Square Roots

Simplifying square roots has several real-world applications, including:

Calculating Distances

When calculating distances, we often need to use square roots to find the length of a side of a right triangle.

Finding Areas of Shapes

When finding the areas of shapes, we often need to use square roots to calculate the area of a square or a rectangle.

Solving Equations

When solving equations, we often need to use square roots to find the solutions to the equation.

Conclusion

In this article, we simplified the expression $\sqrt{36+64}$ using basic algebraic operations and properties of square roots. We also discussed the properties of square roots and provided examples of simplifying square roots using these properties. Finally, we discussed the real-world applications of simplifying square roots.

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Introduction

In our previous article, we simplified the expression $\sqrt{36+64}$ using basic algebraic operations and properties of square roots. In this article, we will answer some frequently asked questions related to simplifying square roots.

Q&A

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, such as 16 or 25. A non-perfect square is a number that cannot be expressed as the square of an integer, such as 41 or 67.

Q: How do I simplify a square root of a sum of two numbers?

A: To simplify a square root of a sum of two numbers, you need to first evaluate the expression inside the square root by adding the two numbers. If the result is a perfect square, you can simplify the square root further.

Q: Can I simplify a square root of a product of two numbers?

A: Yes, you can simplify a square root of a product of two numbers using the property that the square root of a product is the product of the square roots.

Q: What is the square root of a negative number?

A: The square root of a negative number is an imaginary number, which is a complex number that cannot be expressed as a real number.

Q: How do I simplify a square root of a fraction?

A: To simplify a square root of a fraction, you need to first simplify the fraction and then take the square root of the result.

Q: Can I simplify a square root of a decimal number?

A: Yes, you can simplify a square root of a decimal number by first converting the decimal number to a fraction and then simplifying the square root.

Q: What is the square root of 0?

A: The square root of 0 is 0, since 0 multiplied by 0 gives 0.

Q: How do I simplify a square root of a negative fraction?

A: To simplify a square root of a negative fraction, you need to first simplify the fraction and then take the square root of the result.

Q: Can I simplify a square root of a complex number?

A: Yes, you can simplify a square root of a complex number using the properties of complex numbers.

Examples of Simplifying Square Roots

Here are a few examples of simplifying square roots using the properties mentioned above.

Example 1: Simplifying $\sqrt{16+25}$

To simplify $\sqrt{16+25}$, we first evaluate the expression inside the square root by adding 16 and 25.

$\sqrt{16+25} = \sqrt{41}$

Since 41 is not a perfect square, we cannot simplify the square root further.

Example 2: Simplifying $\sqrt{9+16}$

To simplify $\sqrt{9+16}$, we first evaluate the expression inside the square root by adding 9 and 16.

$\sqrt{9+16} = \sqrt{25}$

Since 25 is a perfect square, we can simplify the square root further.

$\sqrt{25} = 5$

Example 3: Simplifying $\sqrt{4+9}$

To simplify $\sqrt{4+9}$, we first evaluate the expression inside the square root by adding 4 and 9.

$\sqrt{4+9} = \sqrt{13}$

Since 13 is not a perfect square, we cannot simplify the square root further.

Real-World Applications of Simplifying Square Roots

Simplifying square roots has several real-world applications, including:

Calculating Distances

When calculating distances, we often need to use square roots to find the length of a side of a right triangle.

Finding Areas of Shapes

When finding the areas of shapes, we often need to use square roots to calculate the area of a square or a rectangle.

Solving Equations

When solving equations, we often need to use square roots to find the solutions to the equation.

Conclusion

In this article, we answered some frequently asked questions related to simplifying square roots. We also provided examples of simplifying square roots using the properties mentioned above. Finally, we discussed the real-world applications of simplifying square roots.