91Ó°ÊÓ

When faced with the problem of multiplying or dividing exponents, it's crucial to simplify the coefficients first. Coefficients are the numerical parts of the terms. In the given exercise, we are dealing with the coefficients \(\frac{1}{5}\) and 25. To simplify, we multiply these coefficients: \[ \frac{1}{5} \times 25 = 5 \] Simply put, multiplying \(\frac{1}{5}\) by 25 is like dividing 25 by 5, which simplifies to 5. Simplifying coefficients makes it easier to handle the variables in the next steps.

Strong grasp of coefficients simplifications paves the way to effortlessly tackle more intricate algebraic expressions.

Combining like terms involves simplifying expressions that have the same base and exponent. In our problem, we have bases 'a' and 'b'. Let's consider each base separately:
- For base 'a', we have \(\frac{a^{4}}{a^{3}}\). When you divide exponents with the same base, you subtract their exponents: \[ a^{4-3} = a^{1} = a \]
- For base 'b', we have \(\frac{b^{4}}{b^{2}}\). Similarly, subtract the exponents: \[ b^{4-2} = b^{2} \]
We combine these to have: \[ a \times b^{2} \].
This process, known as combining like terms, is essential in ensuring you correctly simplify exponents while respecting their algebraic properties.

Understanding the properties of exponents is key to mastering tasks involving multiplication or division of exponents. Here are the core principles applied in our solution:
- **Product of Powers Property**: When you multiply exponents with the same base, you add the exponents: \[ a^m \times a^n = a^{m+n} \]
- **Quotient of Powers Property**: When you divide exponents with the same base, you subtract the exponents: \[ a^m \times \frac{1}{a^n} = a^{m-n} \]
- **Power of a Power Property**: When an exponent is raised to another power, you multiply the exponents: \[ (a^m)^n = a^{mn} \]
In our exercise, we've used these properties to simplify the terms before combining them back. Recognizing and applying these rules will help you decode and simplify even the most complex expressions involving exponents.