Simplify.
$\sqrt{63}$ = ? $\sqrt{}$
Answer (1)
Find the prime factorization of 63: 63 = 9 × 7 .
Rewrite the square root: 63 = 9 × 7 .
Separate the square root: 9 × 7 = 9 × 7 .
Simplify: 63 = 3 7 .
Explanation
Understanding the Problem We are asked to simplify the expression 63 . This means we want to find the largest perfect square that divides 63 and then use the property a ⋅ b = a ⋅ b to simplify the expression.
Finding the Prime Factorization First, we need to find the prime factorization of 63. We can write 63 = 9 × 7 . Since 9 is a perfect square ( 9 = 3 2 ), we can rewrite the expression as 63 = 9 × 7 .
Separating the Square Root Now we use the property of square roots to separate the factors: 9 × 7 = 9 × 7 .
Simplifying the Perfect Square Since 9 = 3 , we can simplify the expression to 3 7 .
Examples
Square roots are used in many areas of math and science. For example, when calculating the distance between two points in a plane, we use the distance formula, which involves square roots. If we have two points ( x 1 , y 1 ) and ( x 2 , y 2 ) , the distance d between them is given by d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . Simplifying square roots helps us to express these distances in their simplest form. Another example is in physics, when calculating the period of a pendulum, which involves a square root. Simplifying the square root makes the calculation easier and more understandable.
