Question

select all expressions that are equivalent to $4^{2} \cdot 4^{8}$. $4^{10}$ $4^{16}$ $\frac{1}{4^{18}}$ $4^{7} \cdot 4^{3}$

Explanation:

Step1: Recall exponent rule for multiplication

When multiplying exponents with the same base, we use the rule \(a^m \cdot a^n = a^{m + n}\). For \(4^{2}\cdot4^{8}\), the base \(a = 4\), \(m = 2\), \(n = 8\). So \(4^{2}\cdot4^{8}=4^{2 + 8}=4^{10}\).

Step2: Analyze each option

• Option \(4^{10}\): From Step 1, we know \(4^{2}\cdot4^{8}=4^{10}\), so this is equivalent.
• Option \(4^{16}\): \(4^{2}\cdot4^{8}=4^{10}

eq4^{16}\), so this is not equivalent.

• Option \(\frac{1}{4^{10}}\): \(\frac{1}{4^{10}}=4^{- 10}

eq4^{10}\), so this is not equivalent.

• Option \(4^{3}\cdot4^{7}\): Using the same exponent rule, \(4^{3}\cdot4^{7}=4^{3 + 7}=4^{10}\), which is equal to \(4^{2}\cdot4^{8}\), so this is equivalent.

Answer:

The equivalent expressions are \(4^{10}\) and \(4^{3}\cdot4^{7}\). So the correct options are:

• \(4^{10}\)
• \(4^{3}\cdot4^{7}\)