To find the value of [tex]\( x \)[/tex] that makes the expression [tex]\(\left(\sqrt[5]{4^3}\right)^x\)[/tex] equal to 4, we will solve the given equation step-by-step.
### Step 1: Simplify the expression inside the parentheses.
Given expression is:
[tex]\[
\left(\sqrt[5]{4^3}\right)^x
\][/tex]
Recall that [tex]\(\sqrt[5]{4^3}\)[/tex] is another way of writing [tex]\(4^{3/5}\)[/tex].
So, the expression becomes:
[tex]\[
(4^{3/5})^x
\][/tex]
### Step 2: Use the power rule for exponents.
When we raise a power to another power, we multiply the exponents. Therefore:
[tex]\[
(4^{3/5})^x = 4^{(3x/5)}
\][/tex]
### Step 3: Set the expression equal to 4.
We need to find [tex]\( x \)[/tex] such that the above expression equals 4:
[tex]\[
4^{(3x/5)} = 4
\][/tex]
### Step 4: Match the exponents.
We notice that the base on both sides of the equation is the same (both are base 4). For the exponents to be equal, we set the exponent on the right side as:
[tex]\[
4^1
\][/tex]
Hence the equation becomes:
[tex]\[
\frac{3x}{5} = 1
\][/tex]
### Step 5: Solve for [tex]\( x \)[/tex].
To find [tex]\( x \)[/tex], solve the equation [tex]\(\frac{3x}{5} = 1\)[/tex]:
[tex]\[
\frac{3x}{5} = 1
\][/tex]
Multiply both sides by 5 to clear the fraction:
[tex]\[
3x = 5
\][/tex]
Next, divide both sides by 3:
[tex]\[
x = \frac{5}{3}
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that makes the original expression equal to 4 is:
[tex]\[
\boxed{\frac{5}{3}}
\][/tex]
