Problem 5 - Algebra 1: Concepts and Skills

Question

Use the product of powers property to write the expression as a single power of the base. \((-5)^{4} \cdot(-5)^{2}\)

Step-by-Step Solution

Answer

The expression \((-5)^4 \cdot(-5)^2\) written as a single power of the base is \((-5)^6\).

Step 1: Identifying the bases and their exponents

Here, the base is -5, one exponent is 4 and the other expoent is 2. The expression can be written like this: \(a^{m} \cdot a^{n}\), where \(a = -5, m = 4\), and \(n = 2\).

Step 2: Applying the Product of Powers Property

According to the product of powers property, when multiplying like bases, the exponents should be added. Therefore, the expression becomes \((-5)^{4+2}\).

Step 3: Adding the exponents

Add together the exponents as follows: 4 + 2 = 6. Therefore, the expression results in \((-5)^6\).

Key Concepts

Exponents

When we talk about exponents, we're referring to the shorthand notation that represents how many times a number, called the base, is multiplied by itself. An expression like \(a^m\) tells us to multiply the base \(a\) by itself \(m\) times.

Understanding exponents is crucial because they are not just numbers; they represent the concept of repeated multiplication, making it simpler to write and work with very large or very small numbers. If we take \(2^3\), it’s the same as saying \(2 \cdot 2 \cdot 2\), which equals \(-8\). It’s a cleaner and more efficient way of expressing multiplication that would otherwise be too lengthy to write out.

Multiplying Like Bases

Multiplying like bases taps into a fundamental property of exponents. When you're multiplying two exponential expressions that have the same base, you simply add their exponents to merge them into a single expression.

For instance, let's consider the base \(10\) raised to the power of \(2\) and \(3\) respectively. Using the product of powers property, we can combine them: \(10^2 \cdot 10^3 = 10^{2+3} = 10^5\). This makes calculations especially efficient in algebra, where you often deal with variables raised to various powers.

Simplifying Expressions

Simplifying expressions is a process of making them easier to understand or solve. It involves combining like terms, factoring, expanding, and applying mathematical properties like the product of powers property.

This skill is vital in algebra as it helps students to transform complex expressions into simpler forms. In doing so, they can solve equations more effectively. Simplification may also include reducing fractions, rationalizing denominators, or canceling common factors. The goal is to rewrite the expression in the simplest form possible without changing its value.