Question
Dahlia is simplifying (3 t) Superscript 4 using these steps:
(3 t) Superscript 4 = 3 t times 3 t times 3 t times 3 t = 3 times 3 times 3 times 3 times t times t times t times t
Although Dahlia is correct so far, which step could she have used instead to simplify the expression (3 t) Superscript 4?
(3 t) Superscript 4 = 3 t Superscript 4
3 Superscript 4 Baseline t = 81 t
4 (3 t) = 12 t
3 Superscript 4 Baseline t Superscript 4 Baseline = 81 t Superscript 4
(3 t) Superscript 4 = 3 t times 3 t times 3 t times 3 t = 3 times 3 times 3 times 3 times t times t times t times t
Although Dahlia is correct so far, which step could she have used instead to simplify the expression (3 t) Superscript 4?
(3 t) Superscript 4 = 3 t Superscript 4
3 Superscript 4 Baseline t = 81 t
4 (3 t) = 12 t
3 Superscript 4 Baseline t Superscript 4 Baseline = 81 t Superscript 4
Answers
GPT-4o mini
Dahlia's current expansion of \((3t)^4\) is correct, but there is a more efficient method to simplify it.
Instead of writing out \((3t)^4\) as \(3t \times 3t \times 3t \times 3t\), she could have used the property of exponents, which states that \((ab)^n = a^n \times b^n\). Applying this property, she can write:
\[
(3t)^4 = 3^4 \times t^4.
\]
Now, calculating \(3^4\):
\[
3^4 = 81,
\]
thus the expression simplifies to:
\[
(3t)^4 = 81t^4.
\]
So, the step she could have used instead is:
\((3t)^4 = 3^4 t^4 = 81t^4\).
This option corresponds to the statement:
- \(3^4 t = 81t\) (which is a misrepresentation, but if interpreted correctly as leading to \(81t^4\), it aligns with the efficient method).
The best direct representation from the options would actually be \(3^4 t^4 = 81 t^4\).
Instead of writing out \((3t)^4\) as \(3t \times 3t \times 3t \times 3t\), she could have used the property of exponents, which states that \((ab)^n = a^n \times b^n\). Applying this property, she can write:
\[
(3t)^4 = 3^4 \times t^4.
\]
Now, calculating \(3^4\):
\[
3^4 = 81,
\]
thus the expression simplifies to:
\[
(3t)^4 = 81t^4.
\]
So, the step she could have used instead is:
\((3t)^4 = 3^4 t^4 = 81t^4\).
This option corresponds to the statement:
- \(3^4 t = 81t\) (which is a misrepresentation, but if interpreted correctly as leading to \(81t^4\), it aligns with the efficient method).
The best direct representation from the options would actually be \(3^4 t^4 = 81 t^4\).