Question

which expression is equivalent to ((6^{5} cdot 8^{5})^{3})?

• (14^{15})
• (14^{30})
• (48^{15})
• (48^{30})

Explanation:

Step1: Apply exponent rule for product

First, use the rule \(a^m \cdot b^m=(a\cdot b)^m\) (but here same exponent on different bases, actually first inside the parentheses: \(6^{5}\cdot8^{5}=(6\times8)^{5}\) because \(a^n\cdot b^n=(ab)^n\). So \(6^{5}\cdot8^{5}=(6\times8)^{5}=48^{5}\).

Step2: Apply exponent rule for power of power

Now, we have \((48^{5})^{3}\). Use the rule \((a^m)^n = a^{m\times n}\). So \(m = 5\), \(n=3\), then \(48^{5\times3}=48^{15}\).

Answer:

\(48^{15}\) (corresponding to the option "48¹⁵")