Can u use laws of indices in this question 

Yes, we can use the laws of indices (exponents) to evaluate the limit \(\lim_{{x \to 5}} \frac{{x^7 - 5^7}}{{x - 5}}\) by factoring the numerator. 

We start with the function:

\[
\lim_{{x \to 5}} \frac{{x^7 - 5^7}}{{x - 5}}
\]

Using the difference of powers formula, we can factor \(x^7 - 5^7\). The formula for the difference of powers is:

\[
a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \cdots + ab^{n-2} + b^{n-1})
\]

For \(x^7 - 5^7\), we have:

\[
x^7 - 5^7 = (x - 5)(x^6 + x^5 \cdot 5 + x^4 \cdot 5^2 + x^3 \cdot 5^3 + x^2 \cdot 5^4 + x \cdot 5^5 + 5^6)
\]

So our limit becomes:

\[
\lim_{{x \to 5}} \frac{{(x - 5)(x^6 + x^5 \cdot 5 + x^4 \cdot 5^2 + x^3 \cdot 5^3 + x^2 \cdot 5^4 + x \cdot 5^5 + 5^6)}}{{x - 5}}
\]

We can cancel the \((x - 5)\) term in the numerator and the denominator:

\[
\lim_{{x \to 5}} (x^6 + x^5 \cdot 5 + x^4 \cdot 5^2 + x^3 \cdot 5^3 + x^2 \cdot 5^4 + x \cdot 5^5 + 5^6)
\]

Now, we simply substitute \(x = 5\) into the polynomial:

\[
5^6 + 5^5 \cdot 5 + 5^4 \cdot 5^2 + 5^3 \cdot 5^3 + 5^2 \cdot 5^4 + 5 \cdot 5^5 + 5^6
\]

\[
5^6 + 5^6 + 5^6 + 5^6 + 5^6 + 5^6 + 5^6
\]

There are seven terms, each equal to \(5^6\):

\[
7 \cdot 5^6
\]

Calculating \(5^6\):

\[
5^6 = 15625
\]

Therefore, the limit is:

\[
7 \cdot 15625 = 109375
\]

So the limit is:

\[
\boxed{109375}
\]

Would you like more details or have any questions? Here are some related questions you might be interested in:

1. How do you factor polynomials using the difference of powers formula?
2. What are other useful factorization techniques for solving limits?
3. Can you provide an example of using the sum of powers formula?
4. How do the laws of exponents help in simplifying expressions?
5. What are some common mistakes to avoid when factoring polynomials?

**Tip:** Always look for opportunities to factor expressions when evaluating limits, as this can often simplify the problem and avoid the need for more complex techniques like L'Hôpital's Rule.

Learn how to use laws of indices (exponents) to evaluate the limit lim(x -> 5) (x^7 - 5^7) / (x - 5). Step-by-step solution with detailed explanation and application of the difference of powers formula. Suitable for advanced high school students studying calculus.

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