To find the values of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] that make the equation
[tex]\[
\sqrt{\frac{50 x^6 y^3}{9 x^8}} = \frac{5 y^6 \sqrt{2 y}}{d x}
\][/tex]
true, we will simplify both sides of the equation and compare the expressions.
### Step 1: Simplify the Left Side
The left-hand side of the equation is:
[tex]\[
\sqrt{\frac{50 x^6 y^3}{9 x^8}}
\][/tex]
Simplifying the expression inside the square root:
[tex]\[
\frac{50 x^6 y^3}{9 x^8} = \frac{50 y^3}{9 x^2}
\][/tex]
Now apply the square root:
[tex]\[
\sqrt{\frac{50 y^3}{9 x^2}} = \frac{\sqrt{50 y^3}}{\sqrt{9 x^2}}
\][/tex]
This can be further simplified to:
[tex]\[
\frac{\sqrt{50} \cdot \sqrt{y^3}}{3 x} = \frac{\sqrt{50} \cdot y^{3/2}}{3 x}
\][/tex]
### Step 2: Simplify the Right Side
The right-hand side of the equation is:
[tex]\[
\frac{5 y^6 \sqrt{2 y}}{d x}
\][/tex]
Simplifying the expression inside the numerator:
[tex]\[
5 y^6 \sqrt{2 y} = 5 \cdot \sqrt{2} \cdot y^6 \cdot y^{1/2} = 5 \cdot \sqrt{2} \cdot y^{6 + 1/2} = 5 \cdot \sqrt{2} \cdot y^{13/2}
\][/tex]
So, the right side becomes:
[tex]\[
\frac{5 \sqrt{2} y^{13/2}}{d x}
\][/tex]
### Step 3: Equate the Simplified Forms
Equating the simplified left and right sides:
[tex]\[
\frac{\sqrt{50} \cdot y^{3/2}}{3 x} = \frac{5 \sqrt{2} \cdot y^{13/2}}{d x}
\][/tex]
For the equation to be true, the powers of [tex]\( y \)[/tex] and [tex]\( x \)[/tex] must match on both sides, as well as the coefficients:
- The power of [tex]\( y \)[/tex] in [tex]\(\sqrt{50} \cdot y^{3/2}\)[/tex] is [tex]\( 3/2 \)[/tex].
- The power of [tex]\( y \)[/tex] in [tex]\( 5 \sqrt{2} \cdot y^{13/2}\)[/tex] is [tex]\( 13/2 \)[/tex].
This implies:
[tex]\[
3/2 = 13/2 - 6c
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
3/2 = 13/2 - 6c \implies 6c = 13/2 - 3/2 \implies 6c = 10/2 \implies 6c = 5 \implies c = \frac{5}{6}
\][/tex]
However, none of the given [tex]\( c \)[/tex] values (1 or 2) match [tex]\(\frac{5}{6}\)[/tex].
### Step 4: Check Given Options for [tex]\( c \)[/tex] and [tex]\( d \)[/tex]
The given options are:
1. [tex]\( c=1, d=3 \)[/tex]
2. [tex]\( c=1, d=9 \)[/tex]
3. [tex]\( c=2, d=8 \)[/tex]
4. [tex]\( c=2, d=9 \)[/tex]
Substituting these back into the equation and simplifying would show us if any simplify to True. Given an unsuccessful evaluation for matching coefficients and exponents, none of the provided options correctly satisfy the equation.
Thus, no values of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] from the provided options make the given equation true.
