What Is This Expression In Simplified Form?$\sqrt{20}+3 \sqrt{50}-2 \sqrt{5}$A. $2 \sqrt{5}+5 \sqrt{2}$B. $15 \sqrt{2}$C. $-15 \sqrt{2}+4 \sqrt{5}$D. $-12 \sqrt{50}$

Understanding the Expression

The given expression is a combination of square roots, and our goal is to simplify it. To do this, we need to first understand the properties of square roots and how they can be manipulated. The expression contains three terms: $\sqrt{20}$, $3\sqrt{50}$, and $-2\sqrt{5}$. We will simplify each term separately and then combine them to get the final result.

Simplifying the First Term: $\sqrt{20}$

To simplify $\sqrt{20}$, we need to find the largest perfect square that divides 20. We know that $20 = 4 \times 5$, and since $4$ is a perfect square, we can write $\sqrt{20}$ as $\sqrt{4 \times 5}$. Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{20}$ as $\sqrt{4} \times \sqrt{5}$. Since $\sqrt{4} = 2$, we have $\sqrt{20} = 2\sqrt{5}$.

Simplifying the Second Term: $3\sqrt{50}$

To simplify $3\sqrt{50}$, we need to find the largest perfect square that divides 50. We know that $50 = 25 \times 2$, and since $25$ is a perfect square, we can write $\sqrt{50}$ as $\sqrt{25 \times 2}$. Using the property of square roots that $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$, we can rewrite $\sqrt{50}$ as $\sqrt{25} \times \sqrt{2}$. Since $\sqrt{25} = 5$, we have $\sqrt{50} = 5\sqrt{2}$. Therefore, $3\sqrt{50} = 3 \times 5\sqrt{2} = 15\sqrt{2}$.

Simplifying the Third Term: $-2\sqrt{5}$

The third term is already simplified, so we can move on to combining the simplified terms.

Combining the Simplified Terms

Now that we have simplified each term, we can combine them to get the final result. We have $\sqrt{20} = 2\sqrt{5}$, $3\sqrt{50} = 15\sqrt{2}$, and $-2\sqrt{5}$. To combine these terms, we need to find a common factor. In this case, the common factor is $\sqrt{5}$. We can rewrite the first term as $2\sqrt{5}$, the second term as $15\sqrt{2}$, and the third term as $-2\sqrt{5}$. Now, we can combine the terms by adding and subtracting the coefficients of $\sqrt{5}$. We have $2\sqrt{5} + 15\sqrt{2} - 2\sqrt{5}$. The $\sqrt{5}$ terms cancel each other out, leaving us with $15\sqrt{2}$.

Conclusion

In conclusion, the simplified form of the expression $\sqrt{20}+3 \sqrt{50}-2 \sqrt{5}$ is $15\sqrt{2}$. This is the correct answer among the given options.

Final Answer

The final answer is: $\boxed{15\sqrt{2}}$

Understanding Square Roots

Square roots are a fundamental concept in mathematics, and they can be a bit tricky to understand at first. But don't worry, we're here to help! In this article, we'll answer some common questions about simplifying square roots.

Q: What is a square root?

A: 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.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number inside the square root. You can then rewrite the square root as the product of the square root of the perfect square and the remaining number.

Q: What is a perfect square?

A: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 4 is a perfect square because it can be expressed as 2 multiplied by 2. Similarly, 9 is a perfect square because it can be expressed as 3 multiplied by 3.

Q: How do I find the largest perfect square that divides a number?

A: To find the largest perfect square that divides a number, you need to factor the number into its prime factors. You can then look for pairs of identical prime factors, which will give you the perfect square.

Q: Can I simplify a square root with a variable inside it?

A: Yes, you can simplify a square root with a variable inside it. The process is the same as simplifying a square root with a number inside it. You need to find the largest perfect square that divides the variable expression and then rewrite the square root as the product of the square root of the perfect square and the remaining variable expression.

Q: How do I simplify a square root with a coefficient?

A: To simplify a square root with a coefficient, you need to multiply the coefficient by the square root of the number inside the square root. For example, if you have 3 times the square root of 16, you can simplify it to 3 times 4, which is 12.

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

A: No, you cannot simplify a square root with a negative number inside it. 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 fraction inside it?

A: To simplify a square root with a fraction inside it, you need to find the largest perfect square that divides the numerator and the denominator. You can then rewrite the square root as the product of the square root of the perfect square and the remaining fraction.

Q: Can I simplify a square root with a decimal number inside it?

A: No, you cannot simplify a square root with a decimal number inside it. The square root of a decimal number is a decimal number, and it cannot be simplified in the same way as a real number.

Q: How do I know if a square root can be simplified?

A: You can simplify a square root if the number inside the square root can be expressed as the product of a perfect square and another number. If the number inside the square root cannot be expressed in this way, then it cannot be simplified.

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

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

• Not finding the largest perfect square that divides the number inside the square root
• Not rewriting the square root as the product of the square root of the perfect square and the remaining number
• Not simplifying the square root of a variable expression
• Not multiplying the coefficient by the square root of the number inside the square root

Conclusion

Simplifying square roots can be a bit tricky, but with practice and patience, you'll become a pro in no time! Remember to find the largest perfect square that divides the number inside the square root, rewrite the square root as the product of the square root of the perfect square and the remaining number, and simplify the square root of a variable expression. Happy simplifying!