Simplify.
$\sqrt{\frac{x^7 z^7}{3}} \cdot \frac{\sqrt{27 w^5}}{\sqrt{w^3 x^5 z^7}}$
Assume that all variables represent positive numbers.
Answer (2)
Combine the expression into a single square root.
Simplify the fraction inside the square root by cancelling common factors.
Take the square root of the simplified expression.
The simplified expression is 3 x w .
Explanation
Understanding the Problem We are asked to simplify the expression 3 x 7 z 7 ⋅ w 3 x 5 z 7 27 w 5 . We assume that all variables represent positive numbers.
Combining into a Single Square Root First, let's rewrite the expression as a single square root: 3 x 7 z 7 ⋅ w 3 x 5 z 7 27 w 5 .
Simplifying the Fraction Now, we simplify the fraction inside the square root by cancelling common factors: 3 x 7 z 7 ⋅ w 3 x 5 z 7 27 w 5 = 3 x 5 z 7 w 3 27 x 7 z 7 w 5 = 9 x 7 − 5 z 7 − 7 w 5 − 3 = 9 x 2 w 2 .
Taking the Square Root Finally, we take the square root of the simplified expression: 9 x 2 w 2 = 3 x w .
Final Answer Therefore, the simplified expression is 3 x w .
Examples
Imagine you are calculating the area of a rectangular garden where the length and width are expressed in terms of simplified variables. Simplifying expressions like this helps in making calculations easier and more manageable, especially when dealing with complex algebraic terms. This skill is crucial in various fields, including engineering, physics, and computer graphics, where simplifying equations can lead to more efficient solutions and better understanding of the underlying principles.
The simplified expression for the given mathematical problem is 3 x w . We combined fractions inside the square root and simplified them step-by-step. This process emphasizes the importance of understanding how to manipulate algebraic expressions effectively.
;
