Find the value of the exponent [tex]$c$[/tex] that makes the second expression equivalent to the first expression where [tex]$x \geq 0$[/tex] and [tex]$y \geq 0$[/tex].
[tex]$\begin{array}{l}
\sqrt[4]{324 x^6 y^8}=\sqrt[4]{2^2 \cdot 3^4 \cdot x^2 \cdot x^6 \cdot y^8} \\
c=\square
\end{array}$[/tex]
Answer (1)
Rewrite 324 as a product of its prime factors: 324 = 2 2 ⋅ 3 4 .
Apply the fourth root to each factor: ( 2 2 ) 1/4 ⋅ ( 3 4 ) 1/4 ⋅ ( x 6 ) 1/4 ⋅ ( y 8 ) 1/4 .
Simplify the exponents: 2 2/4 ⋅ 3 4/4 ⋅ x 6/4 ⋅ y 8/4 = 2 1/2 ⋅ 3 ⋅ x 3/2 ⋅ y 2 .
The exponent of y is 2 .
Explanation
Understanding the Problem We are given the expression 4 324 x 6 y 8 and we want to find the exponent c such that when the expression is simplified, we have a term y c .
Prime Factorization First, we can rewrite 324 as a product of its prime factors: 324 = 2 2 ⋅ 3 4 . So, the expression becomes 4 2 2 ⋅ 3 4 ⋅ x 6 ⋅ y 8 .
Applying the Fourth Root Now, we apply the fourth root to each factor: ( 2 2 ) 1/4 ⋅ ( 3 4 ) 1/4 ⋅ ( x 6 ) 1/4 ⋅ ( y 8 ) 1/4 .
Simplifying Exponents Simplifying the exponents, we get: 2 2/4 ⋅ 3 4/4 ⋅ x 6/4 ⋅ y 8/4 = 2 1/2 ⋅ 3 ⋅ x 3/2 ⋅ y 2 .
Finding the Exponent The simplified expression is 3 2 x 3/2 y 2 . We are looking for the exponent of y , which is c = 2 .
Examples
Understanding exponents and roots is crucial in many scientific fields. For instance, in physics, the kinetic energy of an object is related to the square of its velocity ( K E = 2 1 m v 2 ). Similarly, the period of a simple pendulum is related to the square root of its length ( T = 2 π g L ). Simplifying expressions with exponents and roots allows scientists and engineers to make accurate calculations and predictions in various real-world scenarios.
