The problem asks us to find the value of \(n\) in the given equation involving exponents.

The equation is: \(49^n + 49^n + 49^n + 49^n + 49^n + 49^n + 49^n = 7^{2221}\)

Let's simplify the left side of the equation. We have 7 terms of \(49^n\) added together. This can be written as a multiplication:

\(7 \times 49^n\)

So the equation becomes:

\(7 \times 49^n = 7^{2221}\)

To solve for \(n\), we need to have the same base on both sides of the equation. We know that \(49\) can be expressed as a power of \(7\):

\(49 = 7^2\)

Substitute this into the equation:

\(7 \times (7^2)^n = 7^{2221}\)

Now, we use the exponent rule \((a^m)^n = a^{mn}\) to simplify the left side:

\(7 \times 7^{2n} = 7^{2221}\)

Next, we use the exponent rule \(a^m \times a^n = a^{m+n}\) to combine the terms on the left side. Remember that \(7\) is the same as \(7^1\):

\(7^1 \times 7^{2n} = 7^{1+2n}\)

So the equation is now:

\(7^{1+2n} = 7^{2221}\)

When we have an equation where the bases are equal, the exponents must also be equal. Therefore, we can set the exponents equal to each other:

\(1 + 2n = 2221\)

Now, we solve this linear equation for \(n\):

• Subtract 1 from both sides: \(2n = 2221 - 1\)
• Simplify: \(2n = 2220\)
• Divide by 2: \(n = \frac{2220}{2}\)
• Calculate the value of \(n\): \(n = 1110\)

Let's verify the answer:

If \(n = 1110\), then \(49^n = 49^{1110} = (7^2)^{1110} = 7^{2 \times 1110} = 7^{2220}\).

The left side of the original equation is \(7 \times 49^n = 7 \times 7^{2220}\).

Using the rule \(a^m \times a^n = a^{m+n}\), we get \(7^1 \times 7^{2220} = 7^{1+2220} = 7^{2221}\).

This matches the right side of the original equation, \(7^{2221}\). So, the value \(n = 1110\) is correct.

The steps followed were:

1. Simplify the sum on the left side into a product.
2. Convert the base 49 to base 7.
3. Use exponent rules to simplify both sides to have the same base.
4. Equate the exponents.
5. Solve the resulting linear equation for \(n\).

The final answer is \(n = 1110\).