Question

question 1-3
which of these explains why this statement is true $4^2 \times 4^6 = 4^4 \times 4^4$ ?
\bigcirc the base number is 4.
\bigcirc the exponents are all even.
\bigcirc there are 8 fours on each side of the equation.
\bigcirc there are 2 terms on each side of the equation.

Explanation:

To determine why \(4^{2} \times 4^{6} = 4^{4} \times 4^{4}\) is true, we analyze each option:

• Option 1: The base being 4 doesn't explain the equality of the products. Different exponents with the same base can give different results.
• Option 2: Exponents being even doesn't guarantee the products are equal. For example, \(4^2\times4^2 = 4^4\) but \(4^2\times4^6\) vs \(4^4\times4^4\) isn't explained by evenness alone.
• Option 3: Using the exponent rule \(a^m\times a^n=a^{m + n}\), for the left - hand side: \(4^{2}\times4^{6}=4^{2 + 6}=4^{8}\), which means we are multiplying \(4\times4\) (from \(4^2\)) and \(4\times4\times4\times4\times4\times4\) (from \(4^6\)), so in total, we have \(2 + 6=8\) factors of 4. For the right - hand side: \(4^{4}\times4^{4}=4^{4+4}=4^{8}\), and \(4^4\) has 4 factors of 4, so two \(4^4\) terms give \(4 + 4 = 8\) factors of 4. So there are 8 fours on each side.
• Option 4: Both sides have 2 terms (products), but that doesn't explain why the products are equal.

Answer:

There are 8 fours on each side of the equation.