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 (2)
Factor the expression under the square root, identifying perfect squares.
Extract the square roots of the perfect squares: 128 x 8 y 3 z 9 = 64 ⋅ 2 ⋅ x 8 ⋅ y 2 ⋅ y ⋅ z 8 ⋅ z .
Simplify 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 choices.
Simplifying the Expression To simplify the expression, we need to factor out perfect squares from under the square root. We can rewrite the expression as follows: 128 x 8 y 3 z 9 = 2 7 x 8 y 3 z 9 = 2 6 ⋅ 2 ⋅ x 8 ⋅ y 2 ⋅ y ⋅ z 8 ⋅ z = ( 2 3 ) 2 ⋅ 2 ⋅ ( x 4 ) 2 ⋅ y 2 ⋅ ( z 4 ) 2 ⋅ z = 8 2 ⋅ 2 ⋅ ( x 4 ) 2 ⋅ y 2 ⋅ ( z 4 ) 2 ⋅ z = 8 x 4 y z 4 2 yz
Comparing with the Options Now we compare our simplified expression 8 x 4 y z 4 2 yz with the given options:
2 x 2 z 2 8 y 3 z
4 x 2 y z 3 2 x 2
8 x 4 y z 4 2 yz
64 x 4 y z 4 2 yz
We can see that the third option, 8 x 4 y z 4 2 yz , matches our simplified expression.
Final Answer Therefore, the expression equivalent to 128 x 8 y 3 z 9 is 8 x 4 y z 4 2 yz .
Examples
Square roots appear in many contexts, such as calculating distances using the Pythagorean theorem. For instance, if you're designing a rectangular garden and want to determine the length of the diagonal path, you would use the formula d = l 2 + w 2 , where d is the diagonal, l is the length, and w is the width. Simplifying square roots, as we did in this problem, helps in expressing these distances in a more manageable form, especially when dealing with large numbers or variables.
The equivalent expression to 128 x 8 y 3 z 9 is 8 x 4 y z 4 2 yz , which matches option C. By factoring and simplifying the original expression under the square root, we derived this result. The process involved identifying perfect squares and extracting their roots.
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