Square root of 24 using perfect squares
The Correct Answer and Explanation is :
To find the square root of 24 using perfect squares, we start by breaking down the number 24 into its prime factors. The prime factorization of 24 is:
[
24 = 2^3 \times 3^1
]
Step 1: Identify Perfect Squares
Perfect squares are numbers that can be expressed as the square of an integer. The perfect squares less than 24 are:
- (1^2 = 1)
- (2^2 = 4)
- (3^2 = 9)
- (4^2 = 16)
The largest perfect square that is less than 24 is 16.
Step 2: Simplifying the Square Root
Using the perfect square 16, we can express the square root of 24 as follows:
[
\sqrt{24} = \sqrt{16 \times \frac{24}{16}} = \sqrt{16 \times 1.5}
]
This simplifies to:
[
\sqrt{24} = \sqrt{16} \times \sqrt{1.5} = 4 \times \sqrt{1.5}
]
Step 3: Approximating (\sqrt{1.5})
To further simplify (\sqrt{1.5}), we can use the fact that:
[
1.5 = \frac{3}{2}
]
Thus:
[
\sqrt{1.5} = \sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}
]
Now, substituting this back into our equation, we have:
[
\sqrt{24} = 4 \times \frac{\sqrt{3}}{\sqrt{2}} = \frac{4\sqrt{3}}{\sqrt{2}}
]
Step 4: Final Simplification
To rationalize the denominator, we multiply the numerator and the denominator by (\sqrt{2}):
[
\sqrt{24} = \frac{4\sqrt{3} \times \sqrt{2}}{2} = \frac{4\sqrt{6}}{2} = 2\sqrt{6}
]
Conclusion
Therefore, the exact value of (\sqrt{24}) can be expressed as:
[
\sqrt{24} = 2\sqrt{6}
]
For a numerical approximation, we can compute ( \sqrt{6} ) to be approximately 2.45, giving us:
[
\sqrt{24} \approx 2 \times 2.45 \approx 4.9
]
Thus, we conclude that (\sqrt{24} = 2\sqrt{6} \approx 4.9). This process highlights the importance of utilizing perfect squares to simplify square roots effectively.