1. $a, b, c, d$, and $N$ are positive whole numbers.
a. If $\sqrt[x]{a^{b} c} c^{d}=a c \sqrt[4]{a^{5} c^{2}}$, what is the relationship between $b$ and $d ?$ Explain how you know.

Step 1 ：First, we rewrite the given equation in terms of exponents: $a^{b/x} c^{1/x} c^{d}=a c a^{5/4} c^{1/2}$

Step 2 ：Next, we simplify the equation by combining like terms: $a^{b/x + 1} c^{1/x + d}=a^{5/4 + 1} c^{1/2 + 1}$

Step 3 ：We then equate the exponents of $a$ and $c$ on both sides of the equation to get two equations: $b/x + 1 = 5/4 + 1$ and $1/x + d = 1/2 + 1$

Step 4 ：Solving the first equation for $b$, we get $b = x(5/4 + 1 - 1) = 5x/4$

Step 5 ：Solving the second equation for $d$, we get $d = 1/2 + 1 - 1/x = 3/2 - 1/x$

Step 6 ：Therefore, the relationship between $b$ and $d$ is given by the equations $b = 5x/4$ and $d = 3/2 - 1/x$