Question
Which statement is true
three to the negative 15 over three to the seventh degree is less than three to the -8×3 to the -nine
Three to the -15 over three to the seventh is greater than three to the -8×3 to the -9
Three to the -15th over three to the seventh degree equals three to the -8×3 to the -9
three to the negative 15 over three to the seventh degree is less than three to the -8×3 to the -nine
Three to the -15 over three to the seventh is greater than three to the -8×3 to the -9
Three to the -15th over three to the seventh degree equals three to the -8×3 to the -9
Answers
GPT-4o mini
To evaluate the statements involving powers of three, we can simplify them using the properties of exponents.
1. **Simplifying \( \frac{3^{-15}}{3^{7}} \)**:
\[
\frac{3^{-15}}{3^{7}} = 3^{-15 - 7} = 3^{-22}
\]
2. **Simplifying \( 3^{-8} \times 3^{-9} \)**:
\[
3^{-8} \times 3^{-9} = 3^{-8 - 9} = 3^{-17}
\]
Now we can compare \( 3^{-22} \) and \( 3^{-17} \).
Since \( 3^{-22} \) has a larger negative exponent than \( 3^{-17} \), it is smaller than \( 3^{-17} \):
\[
3^{-22} < 3^{-17}
\]
So the comparison:
\[
\frac{3^{-15}}{3^{7}} < 3^{-8} \times 3^{-9}
\]
is true.
Therefore, the correct statement is:
**"Three to the negative 15 over three to the seventh degree is less than three to the -8×3 to the -nine."**
1. **Simplifying \( \frac{3^{-15}}{3^{7}} \)**:
\[
\frac{3^{-15}}{3^{7}} = 3^{-15 - 7} = 3^{-22}
\]
2. **Simplifying \( 3^{-8} \times 3^{-9} \)**:
\[
3^{-8} \times 3^{-9} = 3^{-8 - 9} = 3^{-17}
\]
Now we can compare \( 3^{-22} \) and \( 3^{-17} \).
Since \( 3^{-22} \) has a larger negative exponent than \( 3^{-17} \), it is smaller than \( 3^{-17} \):
\[
3^{-22} < 3^{-17}
\]
So the comparison:
\[
\frac{3^{-15}}{3^{7}} < 3^{-8} \times 3^{-9}
\]
is true.
Therefore, the correct statement is:
**"Three to the negative 15 over three to the seventh degree is less than three to the -8×3 to the -nine."**