Question

which expression is equivalent to $4^{-6} \times 4^{-2}$?

answer

$\bigcirc$ $4^{12}$ $\bigcirc$ $\frac{1}{4^7}$

$\bigcirc$ $4^4$ $\bigcirc$ $\frac{1}{4^8}$

Explanation:

Step1: Recall the exponent rule for multiplication

When multiplying two exponential expressions with the same base, we add the exponents. The rule is \(a^m\times a^n = a^{m + n}\), where \(a\) is the base and \(m\), \(n\) are the exponents. Here, the base \(a = 4\), \(m=-6\) and \(n = - 2\).
So, \(4^{-6}\times4^{-2}=4^{-6+( - 2)}\)

Step2: Simplify the exponent

Calculate \(-6+( - 2)\). Adding two negative numbers: \(-6-2=-8\). So the expression becomes \(4^{-8}\)

Step3: Recall the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) for any non - zero number \(a\) and positive integer \(n\). Using this rule with \(a = 4\) and \(n = 8\), we get \(4^{-8}=\frac{1}{4^{8}}\)

Answer:

\(\frac{1}{4^{8}}\) (the option \(\frac{1}{4^{8}}\))