Sure! Let's solve this problem step by step.
We are given the expression [tex]\(\frac{2x \cdot 2y^3}{5z^2}\)[/tex] and the values [tex]\(x = 6\)[/tex], [tex]\(y = -4\)[/tex], and [tex]\(z = 14\)[/tex].
1. Substitute the given values into the expression:
[tex]\[
\frac{2 \cdot 6 \cdot 2 \cdot (-4)^3}{5 \cdot 14^2}
\][/tex]
2. Calculate [tex]\(y^3\)[/tex]:
[tex]\[
(-4)^3 = (-4) \times (-4) \times (-4) = -64
\][/tex]
3. Substitute the value of [tex]\(y^3\)[/tex] back into the expression:
[tex]\[
\frac{2 \cdot 6 \cdot 2 \cdot (-64)}{5 \cdot 14^2}
\][/tex]
4. Calculate the product in the numerator:
[tex]\[
2 \cdot 6 = 12
\][/tex]
[tex]\[
12 \cdot 2 = 24
\][/tex]
[tex]\[
24 \cdot (-64) = -1536
\][/tex]
So, the numerator is [tex]\(-1536\)[/tex].
5. Calculate [tex]\(z^2\)[/tex]:
[tex]\[
14^2 = 14 \times 14 = 196
\][/tex]
6. Calculate the product in the denominator:
[tex]\[
5 \cdot 196 = 980
\][/tex]
So, the denominator is 980.
7. Finally, divide the numerator by the denominator to find the result:
[tex]\[
\frac{-1536}{980} \approx -1.5673469387755101
\][/tex]
So, the evaluated expression is [tex]\(\frac{2x \cdot 2y^3}{5z^2} = -1.5673469387755101\)[/tex], with the numerator being [tex]\(-1536\)[/tex] and the denominator being [tex]\(980\)[/tex].
