Which expression is equivalent to $\sqrt{8 x^7 y^8}$? Assume $x \geq 0$.
A. $x y^2 \sqrt{8 x^3}$
B. $2 x^2 y^2 \sqrt{x y^2}$
C. $2 x^3 y^4 \sqrt{2 x}$
D. $4 x^3 y^4 \sqrt{x}$
Answer (2)
The expression 8 x 7 y 8 simplifies to 2 x 3 y 4 2 x , which corresponds to option C. Therefore, the answer is option C. This process involves factoring out perfect squares and simplifying the square root.
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Rewrite the expression: 8 x 7 y 8 = 2 3 x 7 y 8 .
Factor out perfect squares: 2 2 ⋅ 2 ⋅ x 6 ⋅ x ⋅ y 8 .
Simplify the square root: 2 x 3 y 4 2 x .
The equivalent expression is: 2 x 3 y 4 2 x .
Explanation
Understanding the Problem We are asked to find an expression equivalent to 8 x 7 y 8 , assuming that x ≥ 0 . This involves simplifying the given square root expression by factoring out perfect squares.
Factoring the Expression First, we can rewrite the expression as follows: 8 x 7 y 8 = 2 3 x 7 y 8 = 2 2 ⋅ 2 ⋅ x 6 ⋅ x ⋅ y 8 Here, we have factored the numbers and variables inside the square root to identify perfect squares.
Simplifying the Square Root Next, we take the square root of the perfect square factors: 2 2 x 6 y 8 ⋅ 2 x = 2 ⋅ x 6/2 ⋅ y 8/2 ⋅ 2 x = 2 x 3 y 4 2 x This simplifies the expression by extracting the square roots of 2 2 , x 6 , and y 8 .
Identifying the Correct Option Finally, we compare our simplified expression 2 x 3 y 4 2 x with the given options. We see that it matches the third option.
Examples
Square roots appear in many areas of math and science. For example, when calculating the distance between two points in a plane, we use the distance formula which involves a square root. Simplifying radical expressions like this one makes those calculations easier. Also, in physics, you might encounter square roots when dealing with energy or momentum calculations. Simplifying these expressions helps in understanding the relationships between different physical quantities.
