Question

$3^5$ equals 243. explain how to use that fact to quickly evaluate $3^6$.

Explanation:

Step1: Recall exponent rule

Recall the exponent rule \(a^{m + n}=a^m\times a^n\). For \(3^6\), we can write \(6 = 5+1\), so \(3^6=3^{5 + 1}\).

Step2: Apply the rule

Using the rule \(a^{m + n}=a^m\times a^n\) with \(a = 3\), \(m = 5\) and \(n=1\), we get \(3^{5+1}=3^5\times3^1\).

Step3: Substitute known value

We know that \(3^5 = 243\) and \(3^1=3\). So substitute these values: \(3^5\times3^1=243\times3\).

Step4: Calculate the product

Calculate \(243\times3\). \(243\times3 = 729\).

Answer:

To evaluate \(3^6\) using \(3^5 = 243\), we use the exponent rule \(a^{m + n}=a^m\times a^n\). Rewrite \(3^6\) as \(3^{5+1}\), which is \(3^5\times3^1\). Substituting \(3^5 = 243\) and \(3^1 = 3\), we get \(243\times3=729\). So \(3^6 = 729\).