Select The Correct Answer.What Is The Simplified Form Of 45 \sqrt{45} 45 ?A. 5 \sqrt{5} 5 B. 9 ⋅ 5 \sqrt{9 \cdot 5} 9 ⋅ 5 C. 3 5 3 \sqrt{5} 3 5 D. 5 3 5 \sqrt{3} 5 3
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Understanding Square Roots
In mathematics, 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. Square roots are denoted by the symbol √ and are used to find the value of a number that, when multiplied by itself, gives the original number.
Simplifying Square Roots: A Key Concept
Simplifying square roots is an essential concept in mathematics, particularly in algebra and geometry. It involves expressing a square root in its simplest form, which means expressing it as a product of a perfect square and a square root. For example, the square root of 45 can be simplified as √(9 × 5), where 9 is a perfect square.
The Correct Answer: Simplifying √45
Now, let's focus on the question at hand: what is the simplified form of √45? To simplify √45, we need to find the largest perfect square that divides 45. In this case, the largest perfect square that divides 45 is 9, which is equal to 3^2. Therefore, we can simplify √45 as follows:
√45 = √(9 × 5) = √9 × √5 = 3√5
Why is 3√5 the Correct Answer?
The correct answer is 3√5 because it is the simplest form of √45. When we multiply 3 by √5, we get a value that, when squared, equals 45. This is because (3√5)^2 = 3^2 × (√5)^2 = 9 × 5 = 45.
Why are the Other Options Incorrect?
Now, let's examine the other options and explain why they are incorrect.
Option A: √5
Option A is incorrect because it does not take into account the perfect square 9 that divides 45. Simply taking the square root of 5 does not simplify the expression.
Option B: √(9 × 5)
Option B is incorrect because it does not simplify the expression further. While it is true that √(9 × 5) = √9 × √5, it does not take into account the fact that 9 is a perfect square.
Option D: 5√3
Option D is incorrect because it does not simplify the expression correctly. The correct simplification of √45 is 3√5, not 5√3.
Conclusion
In conclusion, the simplified form of √45 is 3√5. This is because 3 is the largest perfect square that divides 45, and when multiplied by √5, it gives a value that, when squared, equals 45. The other options are incorrect because they do not simplify the expression correctly.
Frequently Asked Questions
Q: What is the simplified form of √45?
A: The simplified form of √45 is 3√5.
Q: Why is 3√5 the correct answer?
A: 3√5 is the correct answer because it is the simplest form of √45. When we multiply 3 by √5, we get a value that, when squared, equals 45.
Q: Why are the other options incorrect?
A: The other options are incorrect because they do not simplify the expression correctly. Option A does not take into account the perfect square 9 that divides 45, while Option B does not simplify the expression further. Option D is incorrect because it does not simplify the expression correctly.
Additional Resources
For more information on simplifying square roots, check out the following resources:
- Khan Academy: Simplifying Square Roots
- Mathway: Simplifying Square Roots
- Wolfram Alpha: Simplifying Square Roots
Final Thoughts
Simplifying square roots is an essential concept in mathematics, and it requires a deep understanding of perfect squares and square roots. By following the steps outlined in this article, you can simplify square roots with ease and become a math whiz. Remember, practice makes perfect, so be sure to practice simplifying square roots regularly to become proficient.
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Frequently Asked Questions
Q: What is the simplified form of √16?
A: The simplified form 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. Then, you can express the square root as a product of the perfect square and the square root of the remaining number.
Q: What is the simplified form of √25?
A: The simplified form of √25 is 5, because 5 multiplied by 5 equals 25.
Q: Can I simplify a square root with a variable?
A: Yes, you can simplify a square root with a variable. For example, √(x^2) can be simplified as x, because x multiplied by x equals x^2.
Q: How do I simplify a square root with a coefficient?
A: To simplify a square root with a coefficient, you need to find the largest perfect square that divides the coefficient. Then, you can express the square root as a product of the perfect square and the square root of the remaining number.
Q: What is the simplified form of √(9x^2)?
A: The simplified form of √(9x^2) is 3x, because 3 multiplied by 3 equals 9, and x multiplied by x equals x^2.
Q: Can I simplify a square root with a negative number?
A: No, you cannot simplify a square root with a negative number. Square roots are only defined for non-negative numbers.
Q: How do I simplify a square root with a fraction?
A: To simplify a square root with a fraction, you need to find the largest perfect square that divides the numerator and the denominator. Then, you can express the square root as a product of the perfect square and the square root of the remaining fraction.
Q: What is the simplified form of √(4/9)?
A: The simplified form of √(4/9) is 2/3, because 2 multiplied by 2 equals 4, and 3 multiplied by 3 equals 9.
Common Mistakes to Avoid
Mistake 1: Not finding the largest perfect square
A: Make sure to find the largest perfect square that divides the number inside the square root.
Mistake 2: Not expressing the square root as a product
A: Make sure to express the square root as a product of the perfect square and the square root of the remaining number.
Mistake 3: Not simplifying the expression correctly
A: Make sure to simplify the expression correctly by finding the largest perfect square that divides the number inside the square root.
Tips and Tricks
Tip 1: Use a calculator to check your answers
A: Use a calculator to check your answers and make sure you are getting the correct simplified form.
Tip 2: Practice, practice, practice
A: Practice simplifying square roots regularly to become proficient.
Tip 3: Break down complex expressions
A: Break down complex expressions into smaller parts and simplify each part separately.
Conclusion
Simplifying square roots is an essential concept in mathematics, and it requires a deep understanding of perfect squares and square roots. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying square roots and tackle even the most complex expressions.
Additional Resources
For more information on simplifying square roots, check out the following resources:
- Khan Academy: Simplifying Square Roots
- Mathway: Simplifying Square Roots
- Wolfram Alpha: Simplifying Square Roots
Final Thoughts
Simplifying square roots is a crucial skill in mathematics, and it requires patience, practice, and persistence. By mastering this skill, you can tackle even the most complex expressions and become a math whiz. Remember to practice regularly and use a calculator to check your answers.