Simplify $\sqrt{80}$.A. $16 \sqrt{5}$ B. $5 \sqrt{4}$ C. $4 \sqrt{5}$ D. $20 \sqrt{4}$
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 a single number or a product of a number and a square root. In this article, we will focus on simplifying the square root of 80, which is a common problem in mathematics.
What is a Square Root?
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 by the symbol √.
Simplifying Square Roots
To simplify a square root, we need to find the largest perfect square that divides the number inside the square root. A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 × 4.
Step 1: Factorize the Number
The first step in simplifying a square root is to factorize the number inside the square root. Factorization involves breaking down a number into its prime factors. For example, the number 80 can be factorized as follows:
80 = 2 × 2 × 2 × 2 × 5
Step 2: Identify Perfect Squares
Once we have factorized the number, we need to identify the perfect squares. In this case, we have four 2s, which can be grouped as two pairs of 2s. This means that we have two perfect squares: 2 × 2 = 4 and 2 × 2 = 4.
Step 3: Simplify the Square Root
Now that we have identified the perfect squares, we can simplify the square root. We can take out the perfect squares from the square root, leaving the remaining factors inside the square root. In this case, we can take out the two pairs of 2s, leaving the factor 5 inside the square root.
√80 = √(2 × 2 × 2 × 2 × 5) = √(4 × 4 × 5) = 4√5
Conclusion
In conclusion, simplifying square roots involves factorizing the number inside the square root, identifying perfect squares, and taking out the perfect squares from the square root. By following these steps, we can simplify the square root of 80 to 4√5.
Answer
The correct answer is C. .
Additional Examples
Here are some additional examples of simplifying square roots:
- √48 = √(2 × 2 × 2 × 2 × 3) = 4√3
- √72 = √(2 × 2 × 2 × 3 × 3) = 6√2
- √96 = √(2 × 2 × 2 × 2 × 2 × 3) = 8√3
Tips and Tricks
Here are some tips and tricks for simplifying square roots:
- Always factorize the number inside the square root.
- Identify perfect squares and take them out of the square root.
- Simplify the remaining factors inside the square root.
- Check your answer by multiplying the simplified square root by itself to ensure that it equals the original number.
Conclusion
Introduction
Simplifying square roots is an essential skill in mathematics, particularly in algebra and geometry. In our previous article, we provided a step-by-step guide on how to simplify square roots. In this article, we will answer some frequently asked questions about simplifying square roots.
Q: What is the difference between a perfect square and a square root?
A: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it can be expressed as 4 × 4. A square root, on the other hand, 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 with a variable?
A: To simplify a square root with a variable, you need to factorize the expression inside the square root and identify perfect squares. For example, if you have √(x^2 + 4x + 4), you can factorize the expression as (x + 2)^2. Then, you can take out the perfect square from the square root, leaving the remaining factors inside the square root.
Q: Can I simplify a square root with a negative number?
A: Yes, you can simplify a square root with a negative number. However, you need to remember that the square root of a negative number is an imaginary number. For example, the square root of -16 is 4i, where i is the imaginary unit.
Q: How do I simplify a square root with a fraction?
A: To simplify a square root with a fraction, you need to rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator. For example, if you have √(1/4), you can rationalize the denominator by multiplying the numerator and denominator by √4, which gives you √1/√4 = 1/2.
Q: Can I simplify a square root with a decimal?
A: Yes, you can simplify a square root with a decimal. However, you need to remember that the square root of a decimal is an irrational number. For example, the square root of 0.16 is approximately 0.4.
Q: How do I simplify a square root with a negative exponent?
A: To simplify a square root with a negative exponent, you need to remember that a negative exponent means taking the reciprocal of the expression. For example, if you have √x^(-2), you can simplify it as 1/√x^2 = 1/x.
Q: Can I simplify a square root with a complex number?
A: Yes, you can simplify a square root with a complex number. However, you need to remember that the square root of a complex number is a complex number. For example, the square root of 3 + 4i is approximately 1.6 + 1.3i.
Conclusion
Simplifying square roots is an essential skill in mathematics, and it involves factorizing the number inside the square root, identifying perfect squares, and taking out the perfect squares from the square root. By following these steps and practicing with additional examples, you can become proficient in simplifying square roots.