Problem 2 - A Graphical Approach to College Algebra

Question

Simplify each expression. Leave answers with exponents. $$(-5)^{2} \cdot(-5)^{6}$$

Step-by-Step Solution

Answer

The simplified expression is \((-5)^8\).

Step 1: Identify the Rule

The given expression is \[(-5)^2 \cdot (-5)^6\].Here, we need to apply the product of powers property which states that \(a^m \cdot a^n = a^{m+n}\).

Step 2: Apply the Rule

Apply the product of powers property to the expression.\[(-5)^2 \cdot (-5)^6 = (-5)^{2+6}\].

Step 3: Simplify the Exponent

Add the exponents together: \[2 + 6 = 8\],which gives us \[(-5)^8\].

Step 4: Final Simplified Form

The expression \((-5)^2 \cdot (-5)^6\) is simplified to \[(-5)^8\].

Key Concepts

Product of Powers Property

When dealing with exponents, understanding the product of powers property is key to simplifying expressions like the one given in the exercise. This property, also known as the law of exponents, helps streamline the multiplication of similar base numbers with different exponents.

Essentially, the product of powers property states that for any base "a" and exponents "m" and "n", the expression \(a^m \cdot a^n\) can be rewritten as \(a^{m+n}\). This means you can add the exponents when the bases are the same and multiply them to create a single exponentiated term.

In the exercise, the given expression is \((-5)^2 \cdot (-5)^6\). The base \((-5)\) remains unchanged, while the exponents \(2\) and \(6\) are added together using this property. That simplifies to \((-5)^{2+6}\) or \((-5)^8\). Understanding this rule helps simplify expressions efficiently.

Simplification

Simplification involves reducing an expression to its simplest form. This is a fundamental process in algebra where you apply different mathematical rules to make an expression more manageable.

When you apply the product of powers property, you directly contribute to simplifying an expression. The initial expression \((-5)^2 \cdot (-5)^6\) consists of two parts with the same base, \((-5)\), raised to different powers. Simplification in this context meant using the product of powers property to merge these two parts into a single term with a summed exponent, \((-5)^8\).

Knowing how and when to simplify expressions can make more complex algebraic manipulations easier, ensuring you deal with fewer terms and clearer forms. It is a skill that greatly supports algebraic problem-solving.

Algebraic Expressions

In algebra, expressions are combinations of numbers, variables, and operations. An expression like \((-5)^2 \cdot (-5)^6\) is a specific type of algebraic expression, involving only numbers and operations of exponentiation.

Algebraic expressions can range from simple to complex, consisting of multiple variables and operators. Understanding how to manipulate and simplify such expressions is a crucial skill.

The given exercise is a straightforward scenario of dealing with numbers, but the principles apply universally to algebraic expressions containing variables. For example, if you had an expression like \(x^2 \cdot x^6\), applying the same product of powers property, it simplifies to \(x^{2+6} = x^8\).

Mastery of these basics in algebraic expressions lays the foundation for more advanced algebraic operations, equations, and problem-solving tasks.