Certainly! Let's solve the given equation step by step:
Given:
[tex]\[ 7^5 \times 7^6 = 7^3 \times 7^k \][/tex]
1. Combine the exponents on the left-hand side:
According to the properties of exponents, when you multiply two expressions with the same base, you add the exponents. Therefore:
[tex]\[ 7^5 \times 7^6 = 7^{5+6} = 7^{11} \][/tex]
2. Simplify the equation:
Now the equation looks like this:
[tex]\[ 7^{11} = 7^3 \times 7^k \][/tex]
3. Combine the exponents on the right-hand side:
Similarly, applying the same property of exponents to the right-hand side:
[tex]\[ 7^3 \times 7^k = 7^{3+k} \][/tex]
So, the equation is now:
[tex]\[ 7^{11} = 7^{3+k} \][/tex]
4. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ 11 = 3 + k \][/tex]
5. Solve for [tex]\(k\)[/tex]:
Isolate [tex]\(k\)[/tex] by subtracting 3 from both sides:
[tex]\[ 11 - 3 = k \][/tex]
[tex]\[ k = 8 \][/tex]
Therefore, the value of [tex]\(k\)[/tex] is:
[tex]\[ k = 8 \][/tex]
