Question

use patterns to simplify the expression. then, use properties of exponents to verify the answer.
$(7^3)^4$
use a property of exponents to rewrite the expression, and simplify the result if possible. select the correct choice and, if necessary, fill in the answer box to complete your choice.
a. $(7^3)^4 = 7^{3 + 4}$, which simplifies to \\(square\\). (type exponential notation with positive exponents.)
b. $(7^3)^4 = 7^{3 \cdot 4}$, which simplifies to \\(square\\). (type exponential notation with positive exponents.)
c. $(7^3)^4 = 7^{3 + 4}$, which cannot be simplified further.
d. $(7^3)^4 = 7^{3 - 4}$, which cannot be simplified further.
e. $(7^3)^4 = 7^{3 - 4}$, which simplifies to \\(square\\). (type exponential notation with positive exponents.)
f. $(7^3)^4 = 7^{3 \cdot 4}$, which cannot be simplified further.

Explanation:

Step1: Apply power of a power rule

The power of a power property states that $(a^m)^n = a^{m \cdot n}$. For $(7^3)^4$, substitute $a=7$, $m=3$, $n=4$.
$$(7^3)^4 = 7^{3 \cdot 4}$$

Step2: Calculate the exponent

Compute the product of the exponents: $3 \cdot 4 = 12$.
$$7^{3 \cdot 4} = 7^{12}$$

Answer:

B. $(7^3)^4 = 7^{3 \cdot 4}$, which simplifies to $7^{12}$. (Type exponential notation with positive exponents.)