Simplify. [tex]\sqrt{8} \cdot \sqrt{6}[/tex]
Answer (1)
Multiply the numbers inside the square roots: 8 \t ⋅ 6 = 48 .
Factor 48 to find a perfect square: 48 = 16 ⋅ 3 .
Simplify the square root: 48 = 16 ⋅ 3 = 16 ⋅ 3 .
The simplified expression is: 4 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 8 ⋅ 6 . To do this, we'll use the property that a ⋅ b = a ⋅ b .
Multiplying the Radicands First, we multiply the numbers inside the square roots: 8 ⋅ 6 = 8 ⋅ 6 = 48 .
Factoring 48 Now, we need to simplify 48 by finding the largest perfect square that divides 48. We can factor 48 as 48 = 16 ⋅ 3 , where 16 is a perfect square ( 16 = 4 2 ).
Simplifying the Square Root We can rewrite the square root as: 48 = 16 ⋅ 3 = 16 ⋅ 3 = 4 3 .
Examples
Understanding how to simplify radicals is useful in many areas, such as physics and engineering, where you might need to calculate distances or areas. For example, if you are designing a square garden with an area of 48 square feet, the length of each side would be 48 feet, which simplifies to 4 3 feet. Knowing how to simplify radicals allows you to express this length in its simplest form.
