Let's explore the multiplication of exponents with the same base to simplify the expression \( 7^8 \cdot 7^{-4} \).
When multiplying exponential expressions that have the same base, you add the exponents. This property can be expressed as:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Here, the base \( a \) is 7, \( m \) is 8, and \( n \) is -4.
Following the property, let's add the exponents:
[tex]\[ 7^8 \cdot 7^{-4} = 7^{8 + (-4)} \][/tex]
Perform the addition in the exponent:
[tex]\[ 8 + (-4) = 4 \][/tex]
Therefore, the expression simplifies to:
[tex]\[ 7^8 \cdot 7^{-4} = 7^4 \][/tex]
Among the given options, we find the correct simplification step-by-step:
[tex]\[ 7^8 \cdot 7^{-4}=7^{8+(-4)}=7^4 \][/tex]
Thus, the correct simplification is:
[tex]\[ 7^8 \cdot 7^{-4} = 7^4 \][/tex]
Additionally, let's calculate the final value of \( 7^4 \):
[tex]\[ 7^4 = 7 \times 7 \times 7 \times 7 = 2401 \][/tex]
So, the simplified exponent is \( 4 \) and the simplified result of the expression \( 7^4 \) is \( 2401 \). Overall, the correct answer is:
[tex]\[ 7^8 \cdot 7^{-4}=7^{8+(-4)}=7^4 \][/tex]
