Problem 32 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(5 x^{2}\right)^{3} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is \(125x^6\).

Step 1: Apply the power rule to both constant and variable with exponent inside the parentheses

Using the power rule \((ab)^n = a^n b^n\), apply the power of 3 (the exponent) to each element inside the parentheses (5 and \(x^2\)). $$ \left(5 x^{2}\right)^{3} = 5^{3} \cdot \left(x^2\right)^3 $$

Step 2: Calculate the power of the constant

Calculate the power of the constant, which is \(5^3\). $$ 5^{3} = 5 \cdot 5 \cdot 5 = 125 $$

Step 3: Apply the power rule to the variable with exponent

Now, apply the power rule to the variable with exponent by multiplying the existing exponent (2) with the new exponent (3). $$ \left(x^2\right)^3 = x^{2 \cdot 3} = x^6 $$

Step 4: Combine the results

Finally, combine the results from Steps 2 and 3 into a simplified expression. $$ 125 \cdot x^6 $$ So, the simplified expression for \(\left(5x^2\right)^3\) is \(125x^6\).

Key Concepts

Simplifying Expressions

Simplifying expressions can seem tricky at first, but it's all about breaking down complex math into easier steps. One of the key strategies is using power rules for exponents. For the expression \( (5x^2)^3 \), you start by using the power rule \((ab)^n = a^n b^n\). This rule tells us that when you have a term with multiple factors inside parentheses raised to a power, you raise each factor to that power separately.

First, apply the power to each element: both the constant 5 and the term \(x^2\). This gives us \(5^3\) and \((x^2)^3\). After raising the constant to the power (\(5^3 = 125\)), you then handle the variable term by multiplying exponents (\((x^2)^3 = x^{6}\)).

Finally, you combine the results to get the simplified expression \(125x^6\). Using this method consistently helps in understanding how expressions are simplified using exponent rules.

Natural Numbers

Understanding natural numbers is crucial when dealing with exponents. Natural numbers are positive integers starting from 1, 2, 3, and so on. They are straightforward and do not include zero or negative numbers.

When working with expressions like \( (5x^2)^3 \), the problem specifies that all exponents are natural numbers, making computations simpler since they involve counting numbers.

Natural numbers ensure that the operation is well-defined, meaning you can count how many times to multiply something. In our example, \(x^2\) involves multiplying \(x\) by itself twice, and \( (x^2)^3 \) means doing it three times in total, resulting in \(x^6\).

Having a clear grasp of what natural numbers signify helps to comprehend the mechanics behind exponent rules more easily.

Variable Exponents

Variable exponents might initially appear complex, but they follow specific rules that make them manageable. In expressions like \( (5x^2)^3 \), the end result's exponent on a variable (here, \(x\)) depends on multiplying together the exponents involved.

Using the power rule \( (a^m)^n = a^{m \cdot n} \), you tackle variable exponents by multiplying their powers. For \( (x^2)^3 \), you multiply 2 by 3, giving \(x^6\). This operation is straightforward once you remember that multiplication is the key operation for combining exponents when a variable term is raised to another power.

Working with variable exponents becomes intuitive once you recognize the systematic nature of exponentiation. Being comfortable with these rules allows for simplifying more intricate algebraic expressions with confidence.