91Ó°ÊÓ

Algebra often involves manipulating expressions with exponents in order to simplify them or to solve equations. One important rule for working with exponents is the quotient rule. The quotient rule states that when you divide two powers with the same base, you subtract the exponents. In mathematical terms, if you have a base number 'a' raised to two different powers 'm' and 'n', the rule is expressed as \( a^m / a^n = a^{m-n} \).

This rule is useful for simplifying expressions where variables and numbers are raised to a power and appear in both the numerator and the denominator. As seen in the exercise, the exponent in the denominator becomes negative when moved to the numerator which is an application of the quotient rule. For example, \( x^{12} \) in the denominator is rewritten as \( x^{-12} \) in the numerator, effectively applying the rule.

Expressions with rational exponents may initially seem intimidating, but they follow rules that make them much more manageable. A rational exponent is an exponent expressed as a fraction, such as \( m/n \). The expression \( a^{m/n} \) can be read as 'the nth root of a raised to the m power', meaning that it's equal to \( \sqrt[n]{a^m} \), or vice versa.

This comes into play when simplifying complex radical expressions. For example, the solution involves rewriting the fourth root of a term as a rational exponent: \( y^7 \) under the fourth root becomes \( y^{7/4} \). Understanding how to use rational exponents can simplify the process of working with roots of any level, not just square or cube roots, thereby expanding one's algebraic toolkit.

Radical expressions involve roots, such as square roots, cube roots, and higher-level roots. The general form of a radical expression is \( \sqrt[n]{a} \), where 'n' is the index of the root and 'a' is the radicand - the number under the root symbol. Simplifying radical expressions often involves several steps, including factoring the radicand, eliminating perfect nth powers, and rationalizing the denominator.

The exercise provides an example of simplifying a radical expression by applying the quotient rule to split the expression into separate roots for each term. This results in an expression that is easier to work with. By understanding both the structure of radical expressions and the rules for manipulating them, including the relationship with rational exponents, students can tackle a variety of problems involving roots with confidence.