Answer :

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]