3. Use your knowledge of the laws of exponents to simplify each of the following expressions. Write your answers in the space provided, with positive exponents. marks) a) \( \left.\frac{\left(3 x^{5}\right)^{2}}{x^{4}} \right\rvert\,= \)

To simplify \( \frac{(3 x^{5})^{2}}{x^{4}} \), start with the numerator. Apply the power of a product rule: \( (ab)^n = a^n b^n \). This gives us \( (3^2)(x^5)^2 = 9x^{10} \). Now, our expression looks like \( \frac{9x^{10}}{x^{4}} \). Using the quotient rule \( \frac{a^m}{a^n} = a^{m-n} \), we subtract the exponents of \( x \): \( x^{10-4} = x^{6} \). So, the final answer is \( 9x^{6} \). Now, let’s have a bit of historical fun! The laws of exponents can be traced back to mathematicians like Euclid and later refined by al-Khwarizmi in the 9th century, who contributed significantly to algebra and made complex concepts more approachable, paving the way for many modern mathematical principles we use today. If you're looking to avoid common pitfalls with exponents, always remember to pay attention to negative exponents—they indicate reciprocals! For instance, \( x^{-n} \) means \( \frac{1}{x^n} \). Misapplying the rules, like forgetting to distribute an exponent over multiple terms, is a rookie mistake. Keep your exponent rules straight, and you'll be simplifying like a pro in no time!