If ${\left(3^4\right)}^x=3^{20}$, find the value of x.

The correct answer is 5.

We are given the exponential equation:

${\left(3^4\right)}^x=3^{20}$

Step 1: Apply the power of a power rule of exponents.

The power of a power rule states that when raising an exponential expression to another power, you multiply the exponents together:

${\left(a^m\right)}^n=a^{m·n}$

Applying this rule to the left side of the equation:

${\left(3^4\right)}^x=3^{4·x}=3^{4x}$

Step 2: Rewrite the equation using the simplified left-hand side.

$3^{4x}=3^{20}$

Step 3: Equate the exponents.

Since the bases on both sides of the equation are equal (both are 3), their exponents must also be equal to each other:

$4x=20$

Step 4: Solve for x.

Divide both sides of the equation by 4:

$x=\frac{20}{4}$

$x=5$

Thus, the value of x is 5.