To determine which choice is equivalent to the expression \(5^{9.969}\), we need to simplify each given option and compare their exponents to the original expression’s exponent.
1. Option A: \(5^9 \cdot 5^{96 / 10} \cdot 5^{9 / 100}\)
- Simplify the exponent:
[tex]\[
9 + \frac{96}{10} + \frac{9}{100} = 9 + 9.6 + 0.09 \approx 18.69
\][/tex]
The exponent is \(18.69\).
2. Option B: \(5^9 + 5^{9 / 10} + 6^{6 / 100}\)
- As this expression involves addition rather than multiplication, it cannot be easily combined into a single exponent form like the others, and we need to compare the terms separately:
[tex]\[
5^9 + 5^{0.9} + 6^{0.06}
\][/tex]
The exponent form does not directly match the exponent \(9.969\).
3. Option C: \(5^{9 + 9 / 10 + 9 / 10 + 6 / 1000}\)
- Simplify the exponent:
[tex]\[
9 + 0.9 + 0.9 + 0.006 = 10.806
\][/tex]
The exponent is \(10.806\).
4. Option D: \(5^9 \cdot 5^{9 / 10} \cdot 5^{6 / 100} \cdot 5^{9 / 1000}\)
- Simplify the exponent:
[tex]\[
9 + \frac{9}{10} + \frac{6}{100} + \frac{9}{1000} = 9 + 0.9 + 0.06 + 0.009 = 9.969
\][/tex]
The exponent is \(9.969\).
Comparing the results of these exponents to the original exponent \(9.969\), we find that Option D has the exponent that matches exactly. Thus,
The correct choice is:
[tex]$[tex]$D. 5^9 \cdot 5^{9 / 10} \cdot 5^{6 / 100} \cdot 5^{9 / 1000}$[/tex]$[/tex]
