Question

in the following equation, what is the value of z?
$11^{z} = \left(11^{3}\
ight)^{3}$
$z = \square$

Explanation:

Step1: Recall exponent rule \((a^m)^n = a^{m\times n}\)

For the right - hand side of the equation \(11^{z}=(11^{3})^{3}\), we apply the power - of - a - power rule. According to the rule \((a^{m})^{n}=a^{m\times n}\), when \(a = 11\), \(m = 3\) and \(n=3\), we have \((11^{3})^{3}=11^{3\times3}\).

Step2: Simplify the exponent

Calculate \(3\times3 = 9\). So, \((11^{3})^{3}=11^{9}\).
Since the equation is \(11^{z}=11^{9}\), and for exponential functions with the same base (\(a^{x}=a^{y}\) implies \(x = y\) when \(a>0,a
eq1\)), here \(a = 11>0,a
eq1\), we can conclude that \(z = 9\).

Answer:

\(9\)