Problem 64 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. $$\sqrt{175}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{175} \) is \(5\sqrt{7}\)

Step 1: Identify Square Numbers

Identify the square numbers that can go into 175. The square numbers less than 175 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144. Of these, the square numbers 1 and 25 can go into 175.

Step 2: Divide the Number

Divide 175 by the square number 25. Doing so gives us 7. Write 175 as \(25 \times 7\).

Step 3: Apply Square Root Property

Next, apply the rule of square roots, which says \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). We have \( \sqrt{175} = \sqrt{25} \times \sqrt{7}\). Here 25 is a perfect square, and its square root is 5.

Step 4: Final Simplification

Perform the multiplication to give the final answer. 5 times the square root of 7 equals \(5\sqrt{7}\). So, \( \sqrt{175} \) simplifies to \(5\sqrt{7}\).