Question

Use the product of powers property to simplify the expression. $$ 3^{2} \cdot 3^{5} $$

Short answer

The simplification of \(3^{2} \cdot 3^{5}\) is \(3^{7}\).

Step-by-step solution

Identify the base and the exponents

The expression presented is \(3^{2} \cdot 3^{5}\). Here, the base of the power is 3 in both \(3^{2}\) and \(3^{5}\). The exponents for these powers are 2 and 5 respectively.

Apply the product of powers property

The product of powers property states that \(a^{m} \cdot a^{n} = a^{m+n}\), where a is the common base and m and n are the exponents. This property states that when multiplying two powers with the same base, you can add the exponents. In this case, \(3^{2} \cdot 3^{5}\) becomes \(3^{2+5}\).

Calculate the new exponent

Add the exponents 2 and 5 to find the exponent in the simplified expression. Doing so gives us \(3^{2+5} = 3^{7}\).