Simplify. Remove all perfect squares from inside the square root. 63
The Correct Answer and Explanation is :
To simplify the square root of 63 and remove all perfect squares from inside the square root, we start by factorizing 63 into its prime factors.
Step 1: Factorize 63
The number 63 can be factored as follows:
[
63 = 9 \times 7
]
Here, 9 is a perfect square since (9 = 3^2).
Step 2: Rewrite the Square Root
We can now rewrite the square root of 63 using the factors we found:
[
\sqrt{63} = \sqrt{9 \times 7}
]
Step 3: Apply the Property of Square Roots
Using the property of square roots that states (\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}), we can separate the square root:
[
\sqrt{63} = \sqrt{9} \times \sqrt{7}
]
Step 4: Simplify the Square Root
Next, we simplify (\sqrt{9}):
[
\sqrt{9} = 3
]
Now substituting back, we get:
[
\sqrt{63} = 3 \times \sqrt{7}
]
Final Answer
Thus, the simplified form of (\sqrt{63}) is:
[
\sqrt{63} = 3\sqrt{7}
]
Conclusion
In this simplification process, we removed the perfect square (9) from inside the square root, leaving us with (3\sqrt{7}). This method of breaking down the number into its prime factors and identifying perfect squares is a common technique in simplifying square roots. By consistently applying these steps, one can simplify other square roots in a similar manner, making it easier to work with radical expressions in mathematics.