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]$\begin{array}{l}
\sqrt[4]{324 x^8 y^8}=\sqrt[4]{2^2 \cdot 3^4 \cdot x^2 \cdot x^6 \cdot y^8} \\
=\square
\end{array}$[/tex]
Answer (1)
Rewrite the expression using exponents: ( 324 x 8 y 8 ) 4 1 .
Substitute 324 = 2 2 ⋅ 3 4 : ( 2 2 ⋅ 3 4 ⋅ x 8 ⋅ y 8 ) 4 1 .
Apply the power of a product rule and simplify exponents: 2 2 1 ⋅ 3 1 ⋅ x 2 ⋅ y 2 .
Rewrite and simplify: 3 x 2 y 2 2 . The final answer is 3 x 2 y 2 2 .
Explanation
Understanding the Problem We are given the expression 4 324 x 8 y 8 which is equivalent to 4 2 2 \tcdot 3 4 \tcdot x 8 y 8 . We are also given that x ≥ 0 and y ≥ 0 . Our goal is to simplify the expression and find an equivalent form.
Rewriting with Exponents First, let's rewrite the expression 4 324 x 8 y 8 as ( 324 x 8 y 8 ) 4 1 . This helps us apply exponent rules more easily.
Substituting the Value of 324 We know that 324 = 2 2 ⋅ 3 4 . Substituting this into our expression, we get ( 2 2 ⋅ 3 4 ⋅ x 8 ⋅ y 8 ) 4 1 .
Applying the Power of a Product Rule Now, we apply the power of a product rule, which states that ( a ⋅ b ) n = a n ⋅ b n . Applying this rule, we get ( 2 2 ) 4 1 ⋅ ( 3 4 ) 4 1 ⋅ ( x 8 ) 4 1 ⋅ ( y 8 ) 4 1 .
Simplifying the Exponents Next, we simplify the exponents using the rule ( a m ) n = a m ⋅ n . This gives us 2 4 2 ⋅ 3 4 4 ⋅ x 4 8 ⋅ y 4 8 .
Simplifying the Fractions Simplifying the fractions in the exponents, we have 2 2 1 ⋅ 3 1 ⋅ x 2 ⋅ y 2 .
Rewriting the Fractional Exponent We can rewrite 2 2 1 as 2 . Therefore, our simplified expression is 3 x 2 y 2 2 .
Final Answer Thus, the simplified expression is 3 x 2 y 2 2 .
Examples
Understanding exponents and roots is crucial in many fields, such as physics and engineering. For example, when calculating the period of a pendulum, you use square roots. If you were designing a clock that relies on a pendulum, accurately calculating the period using these mathematical principles would be essential to ensure the clock keeps correct time. This involves understanding how different factors, like the length of the pendulum, affect the period through the square root function.
