Problem 11 - Algebra 1: Concepts and Skills

Question

Use the power of a quotient property to simplify the expression. $$ \left(\frac{1}{2}\right)^{5} $$

Step-by-Step Solution

Answer

The simplified form of \( \left(\frac{1}{2}\right)^{5} \) is \( \frac{1}{32} \).

Step 1: Identify the Base and the Power

First you need to identify the base and the exponent in the given expression. Here, the base is \(\frac{1}{2}\) and the power or exponent is 5.

Step 2: Apply the Power of a Quotient Rule

The power of a quotient rule asserts that \( \left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}} \). Here, apply this rule to the base and the exponent in the given expression. So, \( \left(\frac{1}{2}\right)^{5} = \frac{1^{5}}{2^{5}} = \frac{1}{32} \).

Step 3: Simplify Further if Needed

The resulting expression is already in its simplest form, so no further simplification is needed. Therefore, the simplified expression is \( \frac{1}{32} \)

Key Concepts

Simplifying Expressions

Solving algebraic expressions often involves simplifying them to make them easier to understand. Simplification is the process of making a complex expression more manageable. It involves applying mathematical rules and properties to reduce expressions to their simplest form. For our exercise, the goal is to use the power of a quotient rule to simplify the given expression to its essence.

• First, understand the components of the expression: the base (the number or variable being raised to a power) and the exponent (the power to which the base is raised).
• Next, apply relevant mathematical properties, such as exponent rules and quotient rules, to simplify the expression.

Each step in the simplification process requires attention and practice. Over time, you’ll become more efficient at identifying which rules to apply and how to streamline complex expressions.

Exponents

Exponents are a fundamental part of algebra and are used to represent repeated multiplication of a number by itself. In the expression that we are simplifying, the exponent indicates how many times the base is used as a factor. For example, in \((\frac{1}{2})^{5}\), the base \(\frac{1}{2}\) is multiplied by itself five times.
Understanding exponents includes learning about different rules and properties:

• The power rule: If you multiply like bases, you add the exponents \(a^m \times a^n = a^{m+n}\).
• The power of a power rule: To find the power of a power, you multiply the exponents \((a^m)^n = a^{m \times n}\).
• The power of a quotient rule: This specific rule means if you have \(\left( \frac{a}{b} \right)^{n}\), it becomes \(\frac{a^n}{b^n}\), spreading the exponent across the numerator and the denominator.

These rules allow you to manipulate and transform expressions, making complex calculations more straightforward.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations that define a specific mathematical relationship. Our task was to simplify an expression using the power of a quotient rule, which is a core component of algebra.
When dealing with algebraic expressions:

• Recognize the individual parts of the expression, such as terms, coefficients, and exponents.
• Apply algebraic rules systematically to simplify, expand, or factor expressions.
• Understand that simplifying an expression does not change its value; it just makes it more convenient to work with.

Mastering these skills is crucial for tackling more advanced algebraic problems. It builds a foundation for understanding equations, inequalities, and functions.