Simplify: $\quad 5 \sqrt{24}$
Answer (1)
Factor the number inside the square root: 24 = 4 × 6 .
Rewrite the expression: 5 24 = 5 4 × 6 .
Simplify the square root: 5 × 4 × 6 = 5 × 2 × 6 .
Multiply the constants: 10 6 .
The simplified expression is 10 6 .
Explanation
Understanding the problem We are asked to simplify the expression $5
\sqrt{24}$ and choose the correct answer from the given options.
Factoring the square root First, we need to simplify the square root. We can factor 24 as 24 = 4 × 6 . Since 4 is a perfect square, we can take its square root.
Simplifying the expression Now we can rewrite the expression as:
5 24 = 5 4 × 6
Since 4 = 2 , we have:
5 4 × 6 = 5 × 2 6
Final simplification Finally, we multiply the constants:
5 × 2 6 = 10 6
So, the simplified expression is 10 6 .
Choosing the correct answer Comparing our simplified expression with the given options, we see that 10 6 is one of the options.
Examples
Square roots appear in many contexts, such as calculating distances using the Pythagorean theorem. For example, if you have a right triangle with legs of length 5 and 5, the length of the hypotenuse is 5 2 + 5 2 = 50 = 25 × 2 = 5 2 . Simplifying square roots helps in finding exact values and understanding geometric relationships.
