Which expression is equivalent to start fraction, 4, to the power minus 5 , divided by, 4, to the power 7 , end fraction, dot, 4, to the power minus 7 , question mark

Question

4
7

4
−5

​
⋅4
−7
?
Answer
Multiple Choice Answers
start fraction, 1, divided by, 4, to the power 21 , end fraction
4
21

1
​

4, to the power 194
19

start fraction, 1, divided by, 4, to the power 19 , end fraction
4
19

1
​

start fraction, 1, divided by, 4, to the power 20 , end fraction
4
20

1
​

Answer 1

To simplify the expression \(\frac{4^{-5}}{4^{7}} \cdot 4^{-7}\), we can use the properties of exponents.

1. Start with the given expression: \[ \frac{4^{-5}}{4^{7}} \cdot 4^{-7} \]

2. Simplify the fraction \(\frac{4^{-5}}{4^{7}}\) using the quotient rule for exponents: \[ \frac{a^m}{a^n} = a^{m-n} \] Here, \(a = 4\), \(m = -5\), and \(n = 7\): \[ \frac{4^{-5}}{4^{7}} = 4^{-5 - 7} = 4^{-12} \]

3. Now, multiply by \(4^{-7}\): \[ 4^{-12} \cdot 4^{-7} \] Again using the product rule for exponents: \[ a^m \cdot a^n = a^{m+n} \] Here, \(m = -12\) and \(n = -7\): \[ 4^{-12} \cdot 4^{-7} = 4^{-12 - 7} = 4^{-19} \]

4. The final expression is \(4^{-19}\). This can also be expressed as: \[ \frac{1}{4^{19}} \]

Now, let's match this with the options provided:

• \(\frac{1}{4^{21}}\)
• \(\frac{1}{4^{19}}\) (which is our result)
• \(\frac{1}{4^{20}}\)

Thus, the correct answer is: \[ \frac{1}{4^{19}} \]