Write $\sqrt{-50}$ In Simplest Radical Form. Answer Attempt 1 Out Of 2: $ \square $

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Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying radical expressions, specifically the expression −50\sqrt{-50}. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. 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.

Simplifying −50\sqrt{-50}

To simplify −50\sqrt{-50}, we need to first understand that the square root of a negative number is an imaginary number. An imaginary number is a complex number that, when squared, gives a negative result. In this case, we can rewrite −50\sqrt{-50} as −1⋅50\sqrt{-1} \cdot \sqrt{50}.

Step 1: Simplify 50\sqrt{50}

To simplify 50\sqrt{50}, we need to find the largest perfect square that divides 50. A perfect square is a number that can be expressed as the square of an integer. In this case, the largest perfect square that divides 50 is 25.

25 is a perfect square because it can be expressed as 5 squared. Therefore, we can rewrite 50\sqrt{50} as 25â‹…2\sqrt{25} \cdot \sqrt{2}.

Step 2: Simplify 25â‹…2\sqrt{25} \cdot \sqrt{2}

Now that we have simplified 50\sqrt{50} to 25â‹…2\sqrt{25} \cdot \sqrt{2}, we can further simplify it by evaluating the square root of 25. As we mentioned earlier, 25 is a perfect square because it can be expressed as 5 squared. Therefore, we can rewrite 25\sqrt{25} as 5.

5 multiplied by 2\sqrt{2} is equal to 52\sqrt{2}.

Step 3: Simplify −1⋅52\sqrt{-1} \cdot 5\sqrt{2}

Now that we have simplified 50\sqrt{50} to 525\sqrt{2}, we can further simplify the expression −50\sqrt{-50} by multiplying it by −1\sqrt{-1}. As we mentioned earlier, −1\sqrt{-1} is an imaginary number that, when squared, gives a negative result. Therefore, we can rewrite −1\sqrt{-1} as i.

i multiplied by 525\sqrt{2} is equal to 5i2\sqrt{2}.

Conclusion

In conclusion, we have simplified the expression −50\sqrt{-50} to 5i2\sqrt{2}. This is the simplest radical form of the expression. We hope that this article has provided a clear explanation of the process of simplifying radical expressions and has helped you to understand the concept of imaginary numbers.

Final Answer

Introduction

In our previous article, we discussed how to simplify radical expressions, specifically the expression −50\sqrt{-50}. We broke down the process into manageable steps and provided a clear explanation of each step. In this article, we will continue to explore the topic of simplifying radical expressions and answer some frequently asked questions.

Q&A

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

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, while an irrational number is a number that cannot be expressed as the ratio of two integers.

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

A: Yes, you can simplify a radical expression with a negative number by rewriting it as the product of the square root of the absolute value of the number and the square root of -1.

Q: How do you simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to find the largest perfect square that divides the variable. Then, you can rewrite the variable as the product of the square root of the perfect square and the remaining variable.

Q: Can you simplify a radical expression with a fraction?

A: Yes, you can simplify a radical expression with a fraction by rewriting it as the product of the square root of the numerator and the square root of the denominator.

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed as a rational or irrational number, while an imaginary number is a complex number that, when squared, gives a negative result.

Q: Can you simplify a radical expression with an imaginary number?

A: Yes, you can simplify a radical expression with an imaginary number by rewriting it as the product of the square root of the absolute value of the number and the square root of -1.

Q: How do you simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, you need to find the largest perfect square that divides each term. Then, you can rewrite each term as the product of the square root of the perfect square and the remaining term.

Q: Can you simplify a radical expression with a negative coefficient?

A: Yes, you can simplify a radical expression with a negative coefficient by rewriting it as the product of the square root of the absolute value of the coefficient and the square root of the number.

Q: How do you simplify a radical expression with a variable and a coefficient?

A: To simplify a radical expression with a variable and a coefficient, you need to find the largest perfect square that divides the variable. Then, you can rewrite the variable as the product of the square root of the perfect square and the remaining variable. Finally, you can multiply the coefficient by the square root of the perfect square.

Conclusion

In conclusion, we have answered some frequently asked questions about simplifying radical expressions. We hope that this article has provided a clear explanation of the process of simplifying radical expressions and has helped you to understand the concept of imaginary numbers.

Final Tips

  • Always start by simplifying the radical expression by finding the largest perfect square that divides the number.
  • Use the product rule to simplify radical expressions with multiple terms.
  • Use the quotient rule to simplify radical expressions with fractions.
  • Be careful when simplifying radical expressions with negative numbers or imaginary numbers.

Final Answer

The final answer is: 5i2\sqrt{2}