Question

Simplify. $$ \left(4^{2}\right)^{4} $$

Short answer

The simplified value of \((4^2)^4\) is 65536.

Step-by-step solution

Understanding the problem

We have a number being raised to a power, and then that result is being raised to another power. The base number is 4, and the two exponents are 2 and 4.

Applying the power of a power rule

We need to multiply the exponents. The power of a power rule states that \((a^m)^n = a^{mn}\). Thus if \(a = 4\), \(m = 2\), and \(n = 4\), then our expression \((4^2)^4\) simplifies to \(4^{2*4}\).

Calculating the new exponent

Multiply 2 by 4 to get a new exponent, which is 8. Thus our simplified expression is now \(4^8\).

Calculate the value of the simplified expression

The last step is to calculate the value of \(4^8\), which equals 65536.