Problem 29 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \frac{x^{4}}{x^{5}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(x^{-1}\).

Step 1 - Identify the like bases and exponents

The base in this case is 'x', and the exponents are '4' in the numerator and '5' in the denominator.

Step 2 - Apply the quotient rule

The quotient rule of exponents states that when you divide like bases, subtract the exponents. So, \(x^{4} / x^{5}\) simplifies to \(x^{4-5}\).

Step 3 - Subtract the exponents

Subtract the exponents: \(4-5=-1\). Therefore, the expression simplifies to \(x^{-1}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra, allowing us to transform complex equations into simpler, more manageable forms. When faced with an expression like \(\frac{x^4}{x^5}\), our goal is to make it easier to work with by reducing it using algebraic rules.

• Recognize the components of the expression: identify the base, which is the same in both the numerator and the denominator.
• Apply relevant algebraic rules like the quotient rule of exponents, which aids in dealing with divisions involving powers.

Breaking down expressions helps in revealing the underlying relationships, making it easier to solve equations or compute values later on. The art of simplifying lies in applying rules correctly to shrink the equation while preserving its original value or meaning.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number. They play a crucial role in mathematics by helping simplify complex operations.

• The base is the number or variable being repeatedly multiplied. In our example, the base is 'x'.
• The exponent indicates how many times the base is multiplied by itself. An exponent of 4 means \(x\times x \times x \times x\).

Exponents follow specific rules, such as the product of powers rule and the quotient rule. These rules allow us to manipulate expressions involving powers efficiently. For negative exponents such as \(x^{-1}\), it represents the reciprocal of the power, effectively switching the position of the base between the numerator and the denominator.

Division of Powers

The division of powers involves using exponent rules to simplify expressions with like bases being divided. The key principle here is the quotient rule of exponents, which states: when dividing like bases, subtract the exponents.

• Identify the base in both parts of the fraction; in our scenario, this is 'x'.
• Subtract the exponent of the denominator from the numerator as per the rule: \(\frac{x^m}{x^n} = x^{m-n}\).
• In the given exercise, \(x^4 / x^5\) simplifies to \(x^{4-5} = x^{-1}\).

Understanding this process allows for easy manipulation and simplification of expressions, leading to a clearer view of what the equation conveys. It's essential to practice these steps consistently for confidence and accuracy in algebraic transformations.