What is 192 in simplest form?
A. 32√//3
B. 3√//8
C. 8√3
D. 2√/48
The Correct Answer and Explanation is :
The correct answer is: C. (8\sqrt{3})
To simplify the number (192) and express it in terms of square roots, we need to factor (192) into its prime factors and identify perfect squares. Here’s the step-by-step breakdown:
Step 1: Prime Factorization of (192)
To begin, we factor (192):
- (192) is even, so we can divide by (2):
- (192 \div 2 = 96)
- (96 \div 2 = 48)
- (48 \div 2 = 24)
- (24 \div 2 = 12)
- (12 \div 2 = 6)
- (6 \div 2 = 3)
This gives us:
[
192 = 2^6 \times 3^1
]
Step 2: Identify Perfect Squares
Next, we can group the factors into perfect squares:
- The largest perfect square from (2^6) is (2^6 = (2^3)^2 = 8^2).
- (3^1) does not form a perfect square, but it can remain under the square root.
Step 3: Simplifying the Square Root
Now we can simplify (\sqrt{192}):
[
\sqrt{192} = \sqrt{2^6 \times 3} = \sqrt{(2^3)^2 \times 3} = 2^3 \sqrt{3} = 8\sqrt{3}
]
Conclusion
Thus, the simplest form of (192) expressed as a square root is:
[
\sqrt{192} = 8\sqrt{3}
]
Answer Choice
So, the correct answer from the options provided is:
C. (8\sqrt{3}).
Additional Explanation
Understanding simplification of square roots is essential in algebra. The process involves recognizing perfect squares among the factors of the number, which can significantly reduce the complexity of square roots. Perfect squares are numbers like (1, 4, 9, 16, 25,) and so forth, which are the squares of integers. By factoring out these squares, we simplify the expression, allowing us to express it in a more manageable form.
For anyone learning algebra or working with square roots, mastering the technique of factorization and simplification helps not only in mathematics but also in applications like geometry, physics, and engineering, where roots often appear in formulas and calculations.