Simplify The Expression:$\sqrt[3]{875}$

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Introduction

In mathematics, simplifying expressions is an essential skill that helps us to solve problems more efficiently and accurately. One of the most common types of simplification is simplifying radicals, which involves expressing a radical in its simplest form. In this article, we will focus on simplifying the expression $\sqrt[3]{875}$.

Understanding the Concept of Radicals

Before we dive into simplifying the expression, let's first understand what radicals are. A radical is a mathematical expression that involves a root of a number. The most common types of radicals are square roots, cube roots, and nth roots. The expression $\sqrt[3]{875}$ is a cube root, which means that we need to find a number that, when cubed, gives us 875.

Breaking Down the Number 875

To simplify the expression $\sqrt[3]{875}$, we need to break down the number 875 into its prime factors. This will help us to identify any perfect cubes that can be extracted from the expression. Let's start by finding the prime factors of 875.

Prime Factorization of 875

The prime factorization of 875 is:

875 = 5 × 5 × 5 × 7

Identifying Perfect Cubes

Now that we have the prime factors of 875, let's identify any perfect cubes that can be extracted from the expression. A perfect cube is a number that can be expressed as the cube of an integer. In this case, we can see that 5 × 5 × 5 is a perfect cube, which is equal to 125.

Simplifying the Expression

Now that we have identified the perfect cube, we can simplify the expression $\sqrt[3]{875}$ by extracting the perfect cube from the expression. This gives us:

$\sqrt[3]{875}$ = $\sqrt[3]{5 × 5 × 5 × 7}$

= $\sqrt[3]{125 × 7}$

= $\sqrt[3]{125}$ × $\sqrt[3]{7}$

= 5 × $\sqrt[3]{7}$

Conclusion

In this article, we simplified the expression $\sqrt[3]{875}$ by breaking down the number 875 into its prime factors and identifying any perfect cubes that can be extracted from the expression. We found that 875 can be expressed as 5 × 5 × 5 × 7, and that 5 × 5 × 5 is a perfect cube equal to 125. By extracting the perfect cube from the expression, we were able to simplify the expression to 5 × $\sqrt[3]{7}$.

Final Answer

The final answer to the expression $\sqrt[3]{875}$ is 5 × $\sqrt[3]{7}$.

Additional Tips and Tricks

• When simplifying radicals, it's essential to break down the number into its prime factors to identify any perfect cubes that can be extracted from the expression.
• A perfect cube is a number that can be expressed as the cube of an integer.
• When extracting a perfect cube from a radical expression, make sure to multiply the cube root of the remaining factor by the cube root of the perfect cube.

Common Mistakes to Avoid

• When simplifying radicals, it's easy to get confused and extract the wrong perfect cube. Make sure to carefully identify the perfect cube and extract it correctly.
• When multiplying radicals, make sure to multiply the cube roots of the factors, not the factors themselves.

Real-World Applications

Simplifying radicals has many real-world applications in mathematics and science. For example, in physics, radicals are used to describe the motion of objects in three-dimensional space. In engineering, radicals are used to describe the stress and strain on materials. In finance, radicals are used to describe the growth and decay of investments.

Conclusion

In conclusion, simplifying the expression $\sqrt[3]{875}$ involves breaking down the number 875 into its prime factors and identifying any perfect cubes that can be extracted from the expression. By extracting the perfect cube from the expression, we were able to simplify the expression to 5 × $\sqrt[3]{7}$. This article provides a step-by-step guide on how to simplify radicals and avoid common mistakes.

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Introduction

In our previous article, we simplified the expression $\sqrt[3]{875}$ by breaking down the number 875 into its prime factors and identifying any perfect cubes that can be extracted from the expression. In this article, we will answer some frequently asked questions about simplifying radicals and provide additional tips and tricks to help you master this skill.

Q&A

Q: What is a radical?

A: A radical is a mathematical expression that involves a root of a number. The most common types of radicals are square roots, cube roots, and nth roots.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to break down the number into its prime factors and identify any perfect cubes that can be extracted from the expression. Then, you can extract the perfect cube from the expression and simplify the remaining factor.

Q: What is a perfect cube?

A: A perfect cube is a number that can be expressed as the cube of an integer. For example, 125 is a perfect cube because it can be expressed as 5 × 5 × 5.

Q: How do I identify perfect cubes?

A: To identify perfect cubes, you need to look for numbers that can be expressed as the cube of an integer. You can also use the formula n^3 = n × n × n to help you identify perfect cubes.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. However, you need to remember that the cube root of a negative number is also negative.

Q: Can I simplify a radical expression with a decimal number?

A: Yes, you can simplify a radical expression with a decimal number. However, you need to remember that the cube root of a decimal number is also a decimal number.

Q: How do I multiply radicals?

A: To multiply radicals, you need to multiply the cube roots of the factors. For example, $\sqrt[3]{2} × \sqrt[3]{3}$ = $\sqrt[3]{2 × 3}$ = $\sqrt[3]{6}$.

Q: How do I divide radicals?

A: To divide radicals, you need to divide the cube roots of the factors. For example, $\sqrt[3]{6} ÷ \sqrt[3]{2}$ = $\sqrt[3]{6 ÷ 2}$ = $\sqrt[3]{3}$.

Q: Can I simplify a radical expression with a variable?

A: Yes, you can simplify a radical expression with a variable. However, you need to remember that the cube root of a variable is also a variable.

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

A: To simplify a radical expression with a fraction, you need to simplify the fraction first and then simplify the radical expression.

Additional Tips and Tricks

• When simplifying radicals, it's essential to break down the number into its prime factors to identify any perfect cubes that can be extracted from the expression.
• A perfect cube is a number that can be expressed as the cube of an integer.
• When extracting a perfect cube from a radical expression, make sure to multiply the cube root of the remaining factor by the cube root of the perfect cube.
• When multiplying radicals, make sure to multiply the cube roots of the factors, not the factors themselves.
• When dividing radicals, make sure to divide the cube roots of the factors, not the factors themselves.

Common Mistakes to Avoid

• When simplifying radicals, it's easy to get confused and extract the wrong perfect cube. Make sure to carefully identify the perfect cube and extract it correctly.
• When multiplying radicals, make sure to multiply the cube roots of the factors, not the factors themselves.
• When dividing radicals, make sure to divide the cube roots of the factors, not the factors themselves.

Real-World Applications

Simplifying radicals has many real-world applications in mathematics and science. For example, in physics, radicals are used to describe the motion of objects in three-dimensional space. In engineering, radicals are used to describe the stress and strain on materials. In finance, radicals are used to describe the growth and decay of investments.

Conclusion

In conclusion, simplifying the expression $\sqrt[3]{875}$ involves breaking down the number 875 into its prime factors and identifying any perfect cubes that can be extracted from the expression. By extracting the perfect cube from the expression, we were able to simplify the expression to 5 × $\sqrt[3]{7}$. This article provides a step-by-step guide on how to simplify radicals and avoid common mistakes.