Question

simplify. express your answer using a single exponent.
$(a^6)^0$

Explanation:

Step1: Recall exponent rule

When raising a power to a power, we multiply the exponents. The formula is \((x^m)^n = x^{m\times n}\). Here, \(x = a\), \(m = 6\), and \(n = 0\).

Step2: Apply the rule

Using the rule \((a^6)^0=a^{6\times0}\). Then, \(6\times0 = 0\), so we have \(a^0\). But any non - zero number to the power of 0 is 1. However, if we just follow the exponent simplification (ignoring the base being zero for a moment as the problem is about simplifying the exponent form), we can also note that when we apply the power - of - a - power rule, we get \(a^{6\times0}=a^0\), and if we consider the form with a single exponent, and since \(6\times0 = 0\), the simplified form with a single exponent is \(a^0\). But usually, we know that \(a^0 = 1\) for \(a
eq0\), but the problem says "express your answer using a single exponent", so we can also just present it as \(a^{0}\) or recognize that \(6\times0 = 0\) so \((a^6)^0=a^{0}\). But more precisely, using the power of a power rule: \((a^m)^n=a^{m\times n}\), so here \(m = 6\), \(n = 0\), so \((a^6)^0=a^{6\times0}=a^0\). And since \(a^0 = 1\) (for \(a
eq0\)), but the question is about expressing with a single exponent, so the exponent is 0, so the answer is \(a^0\) (or 1, but as per the exponent form, \(a^0\)). Wait, actually, when simplifying \((a^6)^0\) using the power - of - a - power rule, we multiply the exponents: \(6\times0 = 0\), so \((a^6)^0=a^{0}\). But also, we know that any non - zero number to the 0 power is 1, but if we have to use a single exponent, and considering the general form, the exponent is 0. But maybe the problem expects us to just apply the power - of - a - power rule and get \(a^{6\times0}=a^0\), and since \(a^0\) is a single exponent form, that's the answer. But also, \(a^0 = 1\) for \(a
eq0\), but the problem says "using a single exponent", so the exponent is 0, so the answer is \(a^0\) (or 1, but as an exponent form, \(a^0\)).

Wait, actually, the power - of - a - power rule is \((x^m)^n=x^{m\cdot n}\). So for \((a^6)^0\), we have \(m = 6\), \(n = 0\), so \((a^6)^0=a^{6\times0}=a^0\). So the simplified form with a single exponent is \(a^0\) (or 1, but as per the exponent, it's \(a^0\)).

Answer:

\(a^{0}\) (or 1, but in exponent form, \(a^0\))