Simplify: [tex]$\sqrt{16 x^6}$[/tex]
Answer (1)
Rewrite the expression as a product of square roots: 16 x 6 = 16 ⋅ x 6 .
Simplify the square root of 16: 16 = 4 .
Simplify the square root of x 6 : x 6 = x 3 .
Combine the results to get the simplified expression: 4 x 3 .
Explanation
Understanding the Problem We are asked to simplify the expression 16 x 6 . This involves understanding how square roots and exponents interact. We'll break down the expression into simpler parts and then combine them.
Separating the Square Root First, we can rewrite the expression as a product of square roots: 16 x 6 = 16 ⋅ x 6
Simplifying the Constant Term Now, we simplify each part separately. The square root of 16 is 4, since 4 ⋅ 4 = 16 : 16 = 4
Simplifying the Variable Term Next, we simplify x 6 . Recall that taking the square root of a term with an exponent is the same as raising that term to the power of 2 1 . Therefore: x 6 = ( x 6 ) 2 1 Using the power of a power rule, we multiply the exponents: x 6 = x 6 ⋅ 2 1 = x 3
Combining the Results Now, we combine the simplified parts: 16 x 6 = 4 ⋅ x 3 = 4 x 3 Thus, the simplified expression is 4 x 3 .
Final Answer Therefore, the simplified form of 16 x 6 is 4 x 3 .
Examples
Imagine you are calculating the side length of a square garden whose area is given by 16 x 6 square meters. The side length would be the square root of the area, which simplifies to 4 x 3 meters. This simplification helps you easily determine the dimensions of the garden based on the value of x . Understanding how to simplify such expressions is useful in various fields like engineering, physics, and computer graphics, where you often deal with areas, volumes, and other geometric calculations.
