1b) Simplify 50 \sqrt{50} 50 ​ .

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Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. It involves expressing a square root in its simplest form, which can be achieved by factoring the number inside the square root sign into its prime factors. In this article, we will simplify the square root of 50, which is denoted as $\sqrt{50}$.

Understanding Square Roots

A 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. The square root of a number can be represented mathematically as $\sqrt{x}$, where $x$ is the number inside the square root sign.

Simplifying Square Roots

To simplify a square root, we need to factor the number inside the square root sign into its prime factors. A prime factor is a prime number that divides the number evenly. For example, the prime factors of 12 are 2 and 3, because 2 multiplied by 2 multiplied by 3 equals 12.

Simplifying $\sqrt{50}$

To simplify $\sqrt{50}$, we need to factor 50 into its prime factors. The prime factors of 50 are 2, 5, and 5, because 2 multiplied by 5 multiplied by 5 equals 50. We can write this as:

$$\sqrt{50} = \sqrt{2 \times 5 \times 5}$$

Grouping Prime Factors

Now that we have factored 50 into its prime factors, we can group the prime factors in pairs. In this case, we have two pairs of 5, which can be written as:

$$\sqrt{50} = \sqrt{2 \times 5^2}$$

Simplifying the Square Root

Now that we have grouped the prime factors in pairs, we can simplify the square root. The square root of a number raised to an even power can be simplified by taking the square root of the number and raising it to the power of half the exponent. In this case, we have:

$$\sqrt{50} = \sqrt{2} \times \sqrt{5^2}$$

Evaluating the Square Root

Now that we have simplified the square root, we can evaluate it. The square root of 2 is approximately 1.414, and the square root of 5 squared is 5. Therefore, we can write:

$$\sqrt{50} \approx 1.414 \times 5$$

Conclusion

In conclusion, we have simplified the square root of 50 by factoring it into its prime factors, grouping the prime factors in pairs, and simplifying the square root. The simplified form of $\sqrt{50}$ is $\sqrt{2} \times 5$, which is approximately equal to 7.07.

Final Answer

The final answer is $\boxed{\sqrt{2} \times 5}$.

Additional Tips and Tricks

• When simplifying square roots, always factor the number inside the square root sign into its prime factors.
• Group the prime factors in pairs to simplify the square root.
• The square root of a number raised to an even power can be simplified by taking the square root of the number and raising it to the power of half the exponent.
• Always evaluate the square root to get the final answer.

Common Mistakes to Avoid

• Not factoring the number inside the square root sign into its prime factors.
• Not grouping the prime factors in pairs to simplify the square root.
• Not evaluating the square root to get the final answer.

Real-World Applications

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

• Calculating distances and heights in geometry and trigonometry.
• Solving equations and inequalities in algebra.
• Working with complex numbers in mathematics and engineering.
• Calculating areas and volumes in geometry and calculus.

Conclusion

In conclusion, simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. By factoring the number inside the square root sign into its prime factors, grouping the prime factors in pairs, and simplifying the square root, we can simplify the square root of 50 to $\sqrt{2} \times 5$, which is approximately equal to 7.07.

Introduction

Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we simplified the square root of 50 by factoring it into its prime factors, grouping the prime factors in pairs, and simplifying the square root. In this article, we will answer some frequently asked questions about simplifying square roots.

Q: What is the difference between a square root and a square?

A: A square root 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. A square, on the other hand, is the result of multiplying a number by itself. For example, the square of 4 is 16, because 4 multiplied by 4 equals 16.

Q: How do I simplify a square root?

A: To simplify a square root, you need to factor the number inside the square root sign into its prime factors. Then, group the prime factors in pairs and simplify the square root. For example, to simplify the square root of 50, you would factor 50 into its prime factors (2, 5, and 5), group the prime factors in pairs (2 and 5^2), and simplify the square root (sqrt(2) x 5).

Q: What is the rule for simplifying square roots?

A: The rule for simplifying square roots is to factor the number inside the square root sign into its prime factors, group the prime factors in pairs, and simplify the square root. This rule can be summarized as:

• Factor the number inside the square root sign into its prime factors.
• Group the prime factors in pairs.
• Simplify the square root by taking the square root of the number and raising it to the power of half the exponent.

Q: Can I simplify a square root with a negative number inside the square root sign?

A: No, you cannot simplify a square root with a negative number inside the square root sign. The square root of a negative number is an imaginary number, which cannot be simplified in the same way as a real number.

Q: How do I simplify a square root with a variable inside the square root sign?

A: To simplify a square root with a variable inside the square root sign, you need to factor the variable into its prime factors and then simplify the square root. For example, to simplify the square root of x^2, you would factor x^2 into its prime factors (x x x), group the prime factors in pairs (x^2), and simplify the square root (x).

Q: What is the difference between a perfect square and an imperfect 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 4^2. An imperfect square, on the other hand, is a number that cannot be expressed as the square of an integer. For example, 50 is an imperfect square because it cannot be expressed as the square of an integer.

Q: How do I determine if a number is a perfect square or an imperfect square?

A: To determine if a number is a perfect square or an imperfect square, you need to check if the number can be expressed as the square of an integer. If it can, then it is a perfect square. If it cannot, then it is an imperfect square.

Q: What are some common mistakes to avoid when simplifying square roots?

A: Some common mistakes to avoid when simplifying square roots include:

• Not factoring the number inside the square root sign into its prime factors.
• Not grouping the prime factors in pairs to simplify the square root.
• Not evaluating the square root to get the final answer.
• Not checking if the number is a perfect square or an imperfect square.

Conclusion

In conclusion, simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. By following the rules for simplifying square roots and avoiding common mistakes, you can simplify square roots with ease. Remember to factor the number inside the square root sign into its prime factors, group the prime factors in pairs, and simplify the square root to get the final answer.