Problem 29 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (-5 a)^{3} $$

Step-by-Step Solution

Answer

\(-125a^3\.

Step 1: Identify the Base and the Exponent

The given term is \((-5a)^3\), where \(-5a\) is the base and 3 is the exponent.

Step 2: Apply the Exponent to Each Term in the Base

The exponent of 3 applies to both elements of the base, -5 and \(a\). This rule allows us to rewrite the term as \((-5)^3 * (a)^3\).

Step 3: Calculate the Powers

Following the rule of exponents for real numbers, we calculate each power. \((-5)^3\) is \(-125\) and \(a^3\) remains as it is since \(a\) is a variable.

Step 4: Combine the Results

Combining the calculated powers, we get the final answer: \(-125 * a^3\).

Key Concepts

Powers of Numbers

Understanding the powers of numbers is crucial for mastering algebra. The power, also known as the exponent, tells us how many times to multiply the number, known as the base, by itself. For instance, when we see the expression \( 2^4 \), the base is 2 and the exponent is 4, which means we multiply 2 by itself four times: \( 2 \times 2 \times 2 \times 2 = 16 \).

Exponents can simplify how we represent and solve problems involving multiplication of the same number. Instead of writing \( 2 \times 2 \times 2 \times 2 \), we can easily write \( 2^4 \) which is much cleaner. This is not only efficient but also less error-prone, as manual multiple multiplication increases the chance of making mistakes.

Negative Bases in Exponents

When dealing with negative bases in exponents, the rules change slightly due to the nature of negative numbers. If you come across a term like \((-3)^4\), the negative base is raised to an even exponent, resulting in a positive result because multiplying an even number of negative numbers always yields a positive product. However, if the exponent is odd, as in \((-3)^3\), the result will be negative.

In the context of the given problem, we have \((-5a)^3\), where the base \(-5a\) is negative. Since the exponent 3 is odd, the result of \((-5)^3\) remains negative. It’s essential to keep parenthesis around the negative base when raising it to a power, as the negative sign is part of the base that is being multiplied repeatedly.

Simplifying Algebraic Expressions with Exponents

Simplifying algebraic expressions with exponents may seem daunting, but by understanding a few rules, this process becomes straightforward. To simplify an expression like \((-5a)^3\), each element within the base must be raised to the power of the exponent.

We apply the exponent of 3 to both the numerical coefficient, which is -5 in this case, and the variable \(a\). As discussed earlier, the negative base raised to an odd exponent will produce a negative result, so \((-5)^3 = -125\). We do not evaluate the variable since its value is unknown, so it remains as \(a^3\). Finally, we combine these results to get our simplified expression, \(-125a^3\), which is now in a much more manageable form for further mathematical operations and applications.