Problem 66 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{6 r^{4}}{6 r^{4}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is 1.

Step 1: Identify the base and its exponent

In this case, the base is "r" and the exponent is 4, and they are being multiplied by a constant, which is 6.

Step 2: Apply the quotient rule to simplify exponents of the same base

Since \(r^4\) is in both the numerator and the denominator, we can apply the quotient rule to simplify it. Subtract the exponents: \(r^{4 - 4} = r^0\)

Step 3: Simplify the constants

The constants in the expression are 6 in both the numerator and the denominator. Divide them: \(\frac{6}{6} = 1\)

Step 4: Apply the rule of exponent: If \(a^0 = 1\) for any value of "a" except 0

We know that \(r^0 = 1\) since the base "r" is nonzero.

Step 5: Simplify the expression

Combine the constants and the exponent expression we simplified: \(1 \times 1 = 1\) The simplified expression is 1.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is a fundamental skill that helps make equations and calculations more manageable. Most simplification processes involve reducing complex, lengthy expressions into simpler, more concise forms without changing their values. When working with exponents in algebra, it’s especially important to understand how to simplify expressions that involve powers.

To simplify expressions with exponents, one must familiarize themselves with the basic properties of exponents, such as the product rule, quotient rule, power rule, and others. In the given exercise, the process begins with recognizing that the same base with exponents is present in both the numerator and the denominator. By applying the quotient rule of exponents, which states that you can divide two powers with the same base by subtracting their exponents, the expression simplifies significantly. For instance, in the exercise, we see \( r^{4} / r^{4} \), which simplifies to \( r^{0} \) because 4 minus 4 equals 0. Consequently, any nonzero base raised to the power of 0 is 1, leading to a much simpler expression.

Exponents Algebra

In exponents algebra, numbers are often expressed in exponential form to denote repeated multiplication of a base. This is an essential part of algebra as it provides a shorthand for writing out long multiplications. For example, \( r^{4} \) indicates that you multiply the base \( r \) by itself four times (\( r \times r \times r \times r \)).

Understanding how to manipulate these exponential expressions relies on mastering a set of rules designed to guide algebraic operations involving exponents. These rules make complex problems tractable – like the exercise provided, where both the product rule and quotient rule of exponents are suggested as tools for simplification. The product rule would be used when multiplying two expressions with the same base, by adding their exponents together. On the other hand, as demonstrated in the solution, the quotient rule takes precedence when dividing two expressions with the same base, arriving at the simplified form by subtracting the exponents.

Rules of Exponents

The rules of exponents are a set of guidelines that describe how to handle exponents during multiplication, division, and exponentiation operations. Some of the fundamental rules include the product rule (for multiplying like bases), quotient rule (for dividing like bases), power rule (when raising a power to another power), zero exponent rule (any base to the power of zero is 1), and negative exponent rule (which represents the reciprocal of the base with a positive exponent).

In the exercise provided, the quotient rule is the main focus. This rule states that when dividing like bases, one should subtract the exponent in the denominator from the exponent in the numerator. The solution proceeds by subtracting 4 from 4, resulting in \( r^0 \) which, by the rule of exponents that states any nonzero number raised to the power of zero is 1, simplifies the entire expression down to 1. This is a perfect illustration of how the rules of exponents are applied to simplify what may seem to be a complex expression at first glance. When learning and applying these rules, it is crucial to practice consistently to gain familiarity and to be able to execute the rules efficiently in various problems.