Question

What is the exact solution for 75(1/5)^(x/5)=3

Answer 1

`\large \boxed{x = 10}`

Explanation:

`\begin{array}{rcll}75\left ((1)/(5)\right )^{(x)/(5)} & = & 3& \\\\\left ((1)/(5)\right )^{(x)/(5)} & = & (3)/(75)&\text{Divided each side by 75} \\\\\left ((1)/(5)\right )^{(x)/(5)} & = & (1)/(25)&\text{Simplified} \\\\\left ((1)/(5)\right )^{(x)/(5)} & = &25^(-1)&\text{Applied exponent rule} \\\\\left ((1)/(5)\right )^{(x)/(5)} & = &\left (5^(2) \right )^(-1)&\text{Converted 25 to base 5} \\\\\end{array}`

`\begin{array}{rcll}\left (5^(-1)\right )^{(x)/(5)} & = &\left (5^(2) \right )^(-1)&\text{Applied exponent rule} \\\\5^{-(x)/(5)} & = &5^(-2)&\text{Multiplied exponents} \\\\-(x)/(5) & = &-2 & \text{Equated exponents}\\\\x & = & \mathbf{10} & \text{Multiplied each side by -2}\\\end{array}\\\text{The exact solution is $\large \boxed{\mathbf{x = 10}}$}`