Simplify The Expression: $2 \sqrt{175}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. One of the most common types of expressions that require simplification is those involving square roots. In this article, we will focus on simplifying the expression $2 \sqrt{175}$.

Understanding the Expression

The given expression is $2 \sqrt{175}$. To simplify this expression, we need to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Breaking Down the Expression

To simplify the expression $2 \sqrt{175}$, we need to break it down into its prime factors. The number 175 can be factored as follows:

175=5×5×7175 = 5 \times 5 \times 7

Simplifying the Square Root

Now that we have broken down the number 175 into its prime factors, we can simplify the square root. We can rewrite the expression as follows:

175=5×5×7\sqrt{175} = \sqrt{5 \times 5 \times 7}

Applying the Property of Square Roots

One of the properties of square roots is that we can take the square root of a product by taking the square root of each factor. Using this property, we can rewrite the expression as follows:

5×5×7=5×5×7\sqrt{5 \times 5 \times 7} = \sqrt{5} \times \sqrt{5} \times \sqrt{7}

Simplifying Further

Now that we have applied the property of square roots, we can simplify the expression further. We can rewrite the expression as follows:

5×5×7=5×7\sqrt{5} \times \sqrt{5} \times \sqrt{7} = 5 \times \sqrt{7}

Multiplying by 2

Finally, we need to multiply the simplified expression by 2. This gives us the final simplified expression:

2×5×7=1072 \times 5 \times \sqrt{7} = 10 \sqrt{7}

Conclusion

In conclusion, we have simplified the expression $2 \sqrt{175}$ by breaking it down into its prime factors, applying the property of square roots, and simplifying further. The final simplified expression is $10 \sqrt{7}$.

Example Use Case

Simplifying expressions like $2 \sqrt{175}$ is an essential skill in mathematics. It helps us solve problems efficiently and accurately. For example, in algebra, we often encounter expressions that involve square roots. By simplifying these expressions, we can solve equations and inequalities more easily.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving square roots:

  • Break down the number into its prime factors.
  • Apply the property of square roots to take the square root of each factor.
  • Simplify further by combining like terms.
  • Multiply the simplified expression by any coefficients.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions involving square roots:

  • Failing to break down the number into its prime factors.
  • Not applying the property of square roots correctly.
  • Not simplifying further by combining like terms.
  • Not multiplying the simplified expression by any coefficients.

Final Thoughts

Simplifying expressions involving square roots is an essential skill in mathematics. By breaking down the number into its prime factors, applying the property of square roots, and simplifying further, we can simplify expressions like $2 \sqrt{175}$ and solve problems more efficiently and accurately.

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Introduction

In our previous article, we simplified the expression $2 \sqrt{175}$ by breaking it down into its prime factors, applying the property of square roots, and simplifying further. In this article, we will answer some frequently asked questions related to simplifying expressions involving square roots.

Q&A

Q: What is the property of square roots that we can use to simplify expressions?

A: One of the properties of square roots is that we can take the square root of a product by taking the square root of each factor. This means that we can rewrite the expression $\sqrt{a \times b}$ as $\sqrt{a} \times \sqrt{b}$.

Q: How do we break down a number into its prime factors?

A: To break down a number into its prime factors, we need to find the prime numbers that multiply together to give the original number. For example, the prime factors of 12 are 2 and 3, because 2 multiplied by 3 equals 12.

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. For example, 16 is a perfect square because it can be expressed as 4 squared. A non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do we simplify expressions involving square roots with perfect squares?

A: To simplify expressions involving square roots with perfect squares, we can take the square root of the perfect square and simplify further. For example, the expression $\sqrt{16}$ can be simplified as follows:

16=4×4=4×4=4\sqrt{16} = \sqrt{4 \times 4} = \sqrt{4} \times \sqrt{4} = 4

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

A: A rational number is a number that can be expressed as the ratio of two integers. For example, 3/4 is a rational number. An irrational number is a number that cannot be expressed as the ratio of two integers.

Q: How do we simplify expressions involving square roots with irrational numbers?

A: To simplify expressions involving square roots with irrational numbers, we need to use the property of square roots and simplify further. For example, the expression $\sqrt{2}$ is an irrational number, and it cannot be simplified further.

Q: What is the final simplified expression for $2 \sqrt{175}$?

A: The final simplified expression for $2 \sqrt{175}$ is $10 \sqrt{7}$.

Example Use Cases

Here are some example use cases for simplifying expressions involving square roots:

  • Algebra: Simplifying expressions involving square roots is an essential skill in algebra. It helps us solve equations and inequalities more easily.
  • Geometry: Simplifying expressions involving square roots is also important in geometry. It helps us calculate the area and perimeter of shapes more easily.
  • Trigonometry: Simplifying expressions involving square roots is also important in trigonometry. It helps us calculate the sine, cosine, and tangent of angles more easily.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving square roots:

  • Break down the number into its prime factors.
  • Apply the property of square roots to take the square root of each factor.
  • Simplify further by combining like terms.
  • Multiply the simplified expression by any coefficients.
  • Use the property of square roots to simplify expressions involving perfect squares and irrational numbers.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions involving square roots:

  • Failing to break down the number into its prime factors.
  • Not applying the property of square roots correctly.
  • Not simplifying further by combining like terms.
  • Not multiplying the simplified expression by any coefficients.
  • Not using the property of square roots to simplify expressions involving perfect squares and irrational numbers.

Final Thoughts

Simplifying expressions involving square roots is an essential skill in mathematics. By breaking down the number into its prime factors, applying the property of square roots, and simplifying further, we can simplify expressions like $2 \sqrt{175}$ and solve problems more efficiently and accurately.