Question
which expression is equal to 8 to the power of 6 over 4 to the power of 6
1 over 2 to the power of 6
2 to the power of 6
2 to the power of 0
2
1 over 2 to the power of 6
2 to the power of 6
2 to the power of 0
2
Answers
GPT-4o mini
To simplify the expression \( \frac{8^6}{4^6} \), we can rewrite the bases in terms of powers of 2.
First, note that:
- \( 8 = 2^3 \)
- \( 4 = 2^2 \)
Now we can rewrite the expression:
\[
8^6 = (2^3)^6 = 2^{3 \cdot 6} = 2^{18}
\]
\[
4^6 = (2^2)^6 = 2^{2 \cdot 6} = 2^{12}
\]
Now substitute these back into the original expression:
\[
\frac{8^6}{4^6} = \frac{2^{18}}{2^{12}}
\]
Using the properties of exponents, we can simplify:
\[
\frac{2^{18}}{2^{12}} = 2^{18 - 12} = 2^6
\]
Thus, \( \frac{8^6}{4^6} = 2^6 \).
Therefore, the correct answer is:
**2 to the power of 6**.
First, note that:
- \( 8 = 2^3 \)
- \( 4 = 2^2 \)
Now we can rewrite the expression:
\[
8^6 = (2^3)^6 = 2^{3 \cdot 6} = 2^{18}
\]
\[
4^6 = (2^2)^6 = 2^{2 \cdot 6} = 2^{12}
\]
Now substitute these back into the original expression:
\[
\frac{8^6}{4^6} = \frac{2^{18}}{2^{12}}
\]
Using the properties of exponents, we can simplify:
\[
\frac{2^{18}}{2^{12}} = 2^{18 - 12} = 2^6
\]
Thus, \( \frac{8^6}{4^6} = 2^6 \).
Therefore, the correct answer is:
**2 to the power of 6**.
nebula
correct! now what's the expression equal to 10 to the power of 9 times 10 to the power of -7?
1 over 10 to the power of -2
10 to the power of -63
1 over 10 to the power of -63
1 over 10 to the power of 2
1 over 10 to the power of -2
10 to the power of -63
1 over 10 to the power of -63
1 over 10 to the power of 2
GPT-4o mini
To simplify the expression \( 10^9 \times 10^{-7} \), we can use the properties of exponents. Specifically, when multiplying numbers with the same base, we add the exponents:
\[
10^9 \times 10^{-7} = 10^{9 + (-7)} = 10^{9 - 7} = 10^2
\]
So, \( 10^9 \times 10^{-7} = 10^2 \).
Now, among your options, we can see that \( 10^2 \) is not explicitly listed, but if we want to look at it in an inverse form and express \( 10^2 \) differently with negative exponents, we can rewrite \( 10^2 \) as:
\[
10^2 = \frac{1}{10^{-2}}
\]
However, comparing with the options given, \( 1 \) over \( 10^{-2} \) can be rearranged by multiplying both the numerator and denominator by \( 10^{2} \):
Thus, the correct answer is:
**1 over 10 to the power of -2.**
\[
10^9 \times 10^{-7} = 10^{9 + (-7)} = 10^{9 - 7} = 10^2
\]
So, \( 10^9 \times 10^{-7} = 10^2 \).
Now, among your options, we can see that \( 10^2 \) is not explicitly listed, but if we want to look at it in an inverse form and express \( 10^2 \) differently with negative exponents, we can rewrite \( 10^2 \) as:
\[
10^2 = \frac{1}{10^{-2}}
\]
However, comparing with the options given, \( 1 \) over \( 10^{-2} \) can be rearranged by multiplying both the numerator and denominator by \( 10^{2} \):
Thus, the correct answer is:
**1 over 10 to the power of -2.**
nebula
correct again! great work! now what's the expression equal to (12 to the power of 3) to the power of 4?
1 over 12 to the power of 7
1 over 12 to the power of 12
12 to the power of 7
12 to the power of 12
1 over 12 to the power of 7
1 over 12 to the power of 12
12 to the power of 7
12 to the power of 12
GPT-4o mini
To simplify the expression \( (12^3)^4 \), we can use the power of a power property of exponents, which states that \( (a^m)^n = a^{m \cdot n} \).
Applying this property:
\[
(12^3)^4 = 12^{3 \cdot 4} = 12^{12}
\]
Therefore, \( (12^3)^4 = 12^{12} \).
Among your provided options, the correct answer is:
**12 to the power of 12.**
Applying this property:
\[
(12^3)^4 = 12^{3 \cdot 4} = 12^{12}
\]
Therefore, \( (12^3)^4 = 12^{12} \).
Among your provided options, the correct answer is:
**12 to the power of 12.**
nebula
thanks so much bot! great work! again!!! Which expression is equivalent to 8 to the power of -4 over 4 to the power of -4?
2 to the power of -4
2 to the power of 0
2
1 over 2 to the power of -4
2 to the power of -4
2 to the power of 0
2
1 over 2 to the power of -4
GPT-4o mini
To simplify the expression \( \frac{8^{-4}}{4^{-4}} \), we can rewrite 8 and 4 in terms of base 2:
\[
8 = 2^3 \quad \text{and} \quad 4 = 2^2
\]
Now we can express both terms:
\[
8^{-4} = (2^3)^{-4} = 2^{3 \cdot -4} = 2^{-12}
\]
\[
4^{-4} = (2^2)^{-4} = 2^{2 \cdot -4} = 2^{-8}
\]
Next, we substitute these back into the original expression:
\[
\frac{8^{-4}}{4^{-4}} = \frac{2^{-12}}{2^{-8}}
\]
Using the properties of exponents, we can simplify:
\[
\frac{2^{-12}}{2^{-8}} = 2^{-12 - (-8)} = 2^{-12 + 8} = 2^{-4}
\]
Thus, \( \frac{8^{-4}}{4^{-4}} = 2^{-4} \).
Among your options, the correct answer is:
**2 to the power of -4.**
\[
8 = 2^3 \quad \text{and} \quad 4 = 2^2
\]
Now we can express both terms:
\[
8^{-4} = (2^3)^{-4} = 2^{3 \cdot -4} = 2^{-12}
\]
\[
4^{-4} = (2^2)^{-4} = 2^{2 \cdot -4} = 2^{-8}
\]
Next, we substitute these back into the original expression:
\[
\frac{8^{-4}}{4^{-4}} = \frac{2^{-12}}{2^{-8}}
\]
Using the properties of exponents, we can simplify:
\[
\frac{2^{-12}}{2^{-8}} = 2^{-12 - (-8)} = 2^{-12 + 8} = 2^{-4}
\]
Thus, \( \frac{8^{-4}}{4^{-4}} = 2^{-4} \).
Among your options, the correct answer is:
**2 to the power of -4.**