Problem 12 - Algebra 1: Concepts and Skills

Question

Use the power of a quotient property to simplify the expression. $$ \left(\frac{3}{5}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{27}{125}\).

Step 1: Recognize the Power of a Quotient Property

The expression given \(\left(\frac{3}{5}\right)^{3}\) is a fraction raised to a power. This structure matches the power of a quotient property rule; if \(\left(\frac{a}{b}\right)^{n}\), it equals \(\frac{a^n}{b^n}\).

Step 2: Apply the Power of a Quotient Property

The power of a quotient property indicates that the power applies to both the numerator and denominator. So, \(\left(\frac{3}{5}\right)^{3}\) becomes \(\frac{3^3}{5^3}\).

Step 3: Simplify the Expression

Simplify the expression by evaluating the powers to get the final result: \(\frac{27}{125}\).

Key Concepts

Exponent Rules

Exponent rules are fascinating tools that help simplify mathematical expressions with ease. When dealing with exponents, there are a few key rules you should remember. - If you have a base raised to a power, such as \(a^n\), multiplying \(a\) by itself \(n\) times gives the result.- The multiplication rule: when you multiply like bases, you add the exponents (\(a^m \times a^n = a^{m+n}\)).- The division rule: when dividing like bases, you subtract the exponents (\(a^m \div a^n = a^{m-n}\)).- Power of a power rule: when raising an exponent to another power, you multiply the exponents (\((a^m)^n = a^{m \cdot n}\)). - Last but not least is the power of a quotient property, which states that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). This is particularly useful with fractions raised to a power. Remembering these rules will give you a strong foundation in working with exponents, particularly in simplifying expressions!

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression more straightforward or easier to understand. To simplify means to break down expressions into fewer terms, reducing the complexity. This task uses the rules of arithmetic and algebra to achieve the most compact form of the expression. When simplifying expressions with fractions raised to powers, like \(\left(\frac{3}{5}\right)^3\), the power of a quotient property is beneficial. You first apply the exponent to both the numerator and denominator separately, resulting in \(\frac{3^3}{5^3}\). Then, calculate each exponentiation, resulting in \(\frac{27}{125}\). Here are some general tips:

• Combine like terms.
• Use exponent rules and properties.
• Simplify fractions by dividing the numerator and denominator by their greatest common factor if possible.

This helps in making the expression more manageable for solving or further manipulation.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition, subtraction, multiplication, and division). They are like the building blocks in algebra, and understanding them is crucial. An algebraic expression can be something simple, like \(x + 2\), or more complex, like \((3x^2 + 2x - 5)\). In the context of powers of quotients, such as \(\left(\frac{3}{5}\right)^3\), understanding how variables and constants interact through exponentiation is vital.The goal in working with algebraic expressions often includes:

• Simplifying the expressions by applying algebraic rules and properties.
• Solving equations involving algebraic expressions to find the values of unknown variables.
• Factoring expressions, which involves expressing them as a product of simpler factors.

By grasping how algebraic expressions function and applying exponent rules, simplifying such expressions becomes much more approachable and logical.