Question
Which expression is equivalent to \left(4^{2}\cdot3\right)^{7}
Answers
Answered by
GPT-4o mini
To simplify the expression \((4^{2} \cdot 3)^{7}\), we can apply the power of a product rule which states \((a \cdot b)^{n} = a^{n} \cdot b^{n}\). Therefore, we have:
\[
(4^{2} \cdot 3)^{7} = (4^{2})^{7} \cdot (3)^{7}
\]
Now, we can further simplify \((4^{2})^{7}\) using the power of a power rule, which states \((a^{m})^{n} = a^{m \cdot n}\):
\[
(4^{2})^{7} = 4^{2 \cdot 7} = 4^{14}
\]
Thus, we can rewrite the expression as:
\[
(4^{2} \cdot 3)^{7} = 4^{14} \cdot 3^{7}
\]
So, the expression equivalent to \((4^{2} \cdot 3)^{7}\) is:
\[
4^{14} \cdot 3^{7}
\]
\[
(4^{2} \cdot 3)^{7} = (4^{2})^{7} \cdot (3)^{7}
\]
Now, we can further simplify \((4^{2})^{7}\) using the power of a power rule, which states \((a^{m})^{n} = a^{m \cdot n}\):
\[
(4^{2})^{7} = 4^{2 \cdot 7} = 4^{14}
\]
Thus, we can rewrite the expression as:
\[
(4^{2} \cdot 3)^{7} = 4^{14} \cdot 3^{7}
\]
So, the expression equivalent to \((4^{2} \cdot 3)^{7}\) is:
\[
4^{14} \cdot 3^{7}
\]