Question

use the product property to simplify this expression:
$41^9 \cdot 41^7 \cdot 41^7 = \square$
note: enter your answer as an exponential (with a base and an exponent). do not use your calculator to approximate the number using scientific notation. for example, enter $2^5$ instead of 32.
question help: \\( \boldsymbol{\text{message instructor}} \\)
\\( \boldsymbol{\text{submit}} \\)
\\( \boldsymbol{\text{question 6}} \\)
simplify the expression.
(enter your answer as a whole number or fraction)
$4^{-2} = \square$
question help: \\( \boldsymbol{\text{video}} \\) \\( \boldsymbol{\text{message instructor}} \\)

Explanation:

Step1: Apply product rule for exponents

For $41^9 \cdot 41^7 \cdot 41^7$, use $a^m \cdot a^n = a^{m+n}$.

$$\begin{align*} 41^9 \cdot 41^7 \cdot 41^7 &= 41^{9+7+7} \end{align*}$$

Step2: Calculate the sum of exponents

Add the exponents together.
$9+7+7=23$

Step3: Apply negative exponent rule

For $4^{-2}$, use $a^{-n}=\frac{1}{a^n}$.
$4^{-2}=\frac{1}{4^2}$

Step4: Evaluate the positive exponent

Calculate the value of $4^2$.
$4^2=16$

Answer:

$41^{23}$
$\frac{1}{16}$