Problem 13 - Elementary Algebra

Question

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. $$ \left(\frac{a}{c}\right)^{5} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(\frac{a}{c}\right)^{5}\) using the power rule for quotients. Answer: \(\frac{a^5}{c^5}\)

Step 1: Expression

Make use of the power rule for quotients, the power rule for products, the
power rule for powers, or a combination of these rules to simplify each
expression.
$$
\left(\frac{a}{c}\right)^{5}
$$

Step 2: Apply Rules

Apply appropriate mathematical rules step by step.

Step 3: Result

Question: Simplify the expression \(\left(\frac{a}{c}\right)^{5}\) using the power rule for quotients. Answer: \(\frac{a^5}{c^5}\)

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is an essential skill in algebra. It helps make complex equations more manageable and easier to understand or solve. To simplify an algebraic expression, we often combine like terms, use arithmetic operations, and apply rules of exponents. For instance, the power rule for quotients is used when a fraction is raised to a power, as seen in the exercise \( \left(\frac{a}{c}\right)^{5} \). Here, the power is distributed to both the numerator and the denominator. Such simplification reduces the expression \( \left(\frac{a}{c}\right)^{5} = \frac{a^5}{c^5} \), making each element in the equation more transparent and ready for any further mathematical operations.

Exponentiation

Exponentiation is an arithmetic operation that involves raising a number or expression to a power. The power, or exponent, indicates how many times to multiply the base by itself. Understanding how to work with exponents is critical as it forms the building block for higher-level mathematics. The expression \( \left(\frac{a}{c}\right)^{5} \) represents exponentiation, where the base is \(\frac{a}{c}\) and the exponent is 5. This means \(\frac{a}{c}\) would be multiplied by itself five times. Appropriate manipulation of exponents follows distinct algebraic rules which enable the simplification of complex expressions with ease.

Algebraic Rules of Powers

The algebraic rules of powers, also known as the laws of exponents, include several specific rules which govern how to simplify expressions with exponents. One such rule is the power rule for quotients, which states that a quotient raised to an exponent can be expressed as the numerator and denominator raised to that exponent separately, \( \left(\frac{a}{c}\right)^{n} = \frac{a^n}{c^n} \). This rule was applied in the given exercise to simplify \( \left(\frac{a}{c}\right)^{5} \) into \( \frac{a^5}{c^5} \). Such rules are fundamental to algebra and are frequently used to break down and solve equations that involve exponents. By mastering these rules, students can tackle various complex problems with confidence and precision.