To solve the equation [tex]\( y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} = -11 y \sqrt[3]{6 y} \)[/tex], we need to identify two specific values that will satisfy the equation:
The equation is:
[tex]\[ y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} = -11 y \sqrt[3]{6 y} \][/tex]
We can rearrange this as:
[tex]\[ y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} + 11 y \sqrt[3]{6 y} = 0 \][/tex]
Combining like terms:
[tex]\[ y \sqrt[3]{6 y} + 11 y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} = 0 \][/tex]
[tex]\[ (1+11) y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} = 0 \][/tex]
[tex]\[ 12 y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} = 0 \][/tex]
Here, the terms we need to balance are:
1. The coefficient with [tex]\( y \sqrt[3]{6 y} \)[/tex]
2. The coefficient with [tex]\( \sqrt[3]{48 y^{-1}} \)[/tex]
For the equation to hold true, the coefficients should be balanced correctly, which happens when:
1. The coefficient with [tex]\( \sqrt[3]{48 y^{-1}} \)[/tex] term is 14.
2. The coefficient with [tex]\( y \sqrt[3]{6 y} \)[/tex] term is 11, which means we multiply it with [tex]\(-1\)[/tex], resulting in [tex]\( -11 \)[/tex].
Given the balancing, the values we need are:
[tex]\[ -11 \ \text{and} \ 14 \][/tex]
These satisfy the equation, and hence,
[tex]\[ y \sqrt[3]{6 y} - 14 \sqrt[3]{48 y^{-1}} = -11 y \sqrt[3]{6 y} \][/tex]
So, the correct values will be 14 and -11.
