Which expression is equivalent to $\sqrt{128 x^8 y^3 z^9}$ ? Assume $y \geq 0$ and $z \geq 0$.
A. $2 x^2 z^2 \sqrt{8 y^3 z}$
B. $4 x^2 y z^3 \sqrt{2 x^2}$
C. $8 x^4 y z^4 \sqrt{2 y z}$
D. $64 x^4 y z^4 \sqrt{2 y z}$
Answer (1)
Factor the expression under the square root: 128 x 8 y 3 z 9 = 64 ⋅ 2 ⋅ x 8 ⋅ y 2 ⋅ y ⋅ z 8 ⋅ z .
Simplify the perfect squares: 64 = 8 , x 8 = x 4 , y 2 = y , z 8 = z 4 .
Rewrite the expression: 8 x 4 y z 4 2 yz .
The equivalent expression is: 8 x 4 y z 4 2 yz .
Explanation
Understanding the Problem We are given the expression 128 x 8 y 3 z 9 and the conditions y ≥ 0 and z ≥ 0 . Our goal is to simplify this expression and find an equivalent one among the given options.
Factoring the Expression First, we can break down the number 128 into its prime factors. We know that 128 = 2 7 = 2 6 ⋅ 2 = 64 ⋅ 2 . So, we can rewrite the expression as 64 ⋅ 2 ⋅ x 8 ⋅ y 3 ⋅ z 9 .
Separating the Terms Now, we can separate the terms under the square root: 64 ⋅ 2 ⋅ x 8 ⋅ y 3 ⋅ z 9 = 64 ⋅ 2 ⋅ x 8 ⋅ y 3 ⋅ z 9 .
Simplifying Each Term We simplify each term: 64 = 8 , x 8 = x 4 , y 3 = y 2 ⋅ y = y y , and z 9 = z 8 ⋅ z = z 4 z .
Combining the Terms Putting it all together, we have 8 ⋅ 2 ⋅ x 4 ⋅ y y ⋅ z 4 z = 8 x 4 y z 4 2 yz .
Finding the Matching Option Comparing our simplified expression 8 x 4 y z 4 2 yz with the given options, we find that it matches the third option.
Final Answer Therefore, the equivalent expression is 8 x 4 y z 4 2 yz .
Examples
Square roots appear in many physics and engineering problems, such as calculating the period of a pendulum or determining the speed of an object. Simplifying expressions with square roots helps in making these calculations easier and more accurate. For example, if you are designing a bridge and need to calculate the tension in a cable, you might encounter an expression involving square roots. Simplifying this expression will allow you to determine the required strength of the cable more easily.
