Simplify [tex]$\sqrt{72}$[/tex]
A. [tex]$2 \sqrt{6}$[/tex]
B. [tex]$36 \sqrt{2}$[/tex]
C. [tex]$12 \sqrt{6}$[/tex]
D. [tex]$6 \sqrt{2}$[/tex]
Answer (1)
Find the prime factorization of 72: 72 = 2 3 × 3 2 .
Rewrite the square root: 72 = 2 3 × 3 2 .
Simplify by taking out perfect squares: 2 2 × 2 × 3 2 = 2 × 3 × 2 .
The simplified form is 6 2 .
Explanation
Understanding the problem We need to simplify the square root of 72, which means finding the largest perfect square that divides 72 and then simplifying the radical.
Prime Factorization of 72 Let's find the prime factorization of 72. We can start by dividing 72 by 2, which gives us 36. Then, 36 is 6 times 6, and 6 is 2 times 3. So, the prime factorization of 72 is 2 × 2 × 2 × 3 × 3 , or 2 3 × 3 2 .
Rewriting the Square Root Now we can rewrite the square root of 72 using its prime factorization: 72 = 2 3 × 3 2 .
Simplifying the Square Root To simplify the square root, we look for pairs of identical factors. We have 2 3 , which is 2 2 × 2 , and 3 2 . So we can rewrite the expression as 2 2 × 2 × 3 2 . Taking the square root of the perfect squares, we get 2 × 3 × 2 , which simplifies to 6 2 .
Selecting the Correct Option Comparing our simplified expression, 6 2 , with the given options, we see that it matches option D.
Final Answer Therefore, the simplified form of 72 is 6 2 .
Examples
Square roots are used in many real-world applications, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 6 and 6, the length of the hypotenuse is 6 2 + 6 2 = 36 + 36 = 72 = 6 2 . Simplifying square roots helps in expressing these lengths in a more understandable form. Another example is in physics, where square roots appear in formulas for calculating speed, energy, and other physical quantities. Simplifying radicals makes these calculations easier to handle and interpret.
