Problem 41 - Beginning and Intermediate Algebra

Question

Simplify. $$\left(-2 x^{3}\right)^{5}$$

Step-by-Step Solution

Answer

The simplified expression is \(-32x^{15}\).

Step 1: Identify the rules to apply

In this case, we will apply the rules for exponentiation and for the product of powers. The rules are: 1. \((a^m)^n = a^{mn}\) 2. \((ab)^n = a^n b^n\)

Step 2: Apply the product of powers rule

Rule 2 allows us to distribute the exponent inside the parenthesis. Apply the rule to the expression: \((-2x^3)^5 = (-2)^5 (x^3)^5\)

Step 3: Apply the exponentiation rule

Rule 1 allows us to find the product of the exponents. Apply the rule to the expression: \((-2)^5 = -32\) \((x^3)^5 = x^{3\cdot 5} = x^{15}\)

Step 4: Combine the simplified terms

Multiply the simplified terms to obtain the final simplified expression: \(-32x^{15}\) So, the simplified expression is \(-32x^{15}\).

Key Concepts

Product of Powers

When simplifying expressions with exponents, one useful tool is the product of powers rule. This rule helps when you have a power raised to another power. For instance, in the expression \((-2x^3)^5\), it allows us to manage the exponents multiplying one another.

The rule states that \((a^b)^c = a^{bc}\). So, when you have a power raised to another power, you can multiply the exponents together. It simplifies the calculation immensely:

• Break down the expression.
• Multiply the powers separately.
• Streamline the expression further.

This simplifies our task at hand and is the cornerstone of much easier algebraic manipulation.

Power Rule

The power rule facilitates exponentiation with greater clarity and simplicity. When dealing with expressions like \((-2)^5 \) and \((x^3)^5\), the power rule comes into play directly.

Specifically, the power rule \((a^m)^n = a^{mn}\) states that you multiply the exponents when raising a power to a power. This turns daunting calculations into a quick multiplication task:

• For \((-2)^5\), calculate the base raised to the power directly, resulting in \(-32\).
• For \((x^3)^5\), multiply the exponents, resulting in \((3 \times 5 = 15)\).

This approach speeds up the process by converting complex expressions into simpler forms easily manipulated.

Polynomial Simplification

Simplifying polynomials makes them easier to work with, especially in algebraic equations. After applying the product of powers and the power rule, combining terms efficiently is crucial.

Once we reach \(-32x^{15}\), we gather our simplified pieces into one expression. This is achieved by multiplying the coefficients and combining variable powers:

• Combine \(-32\) and \((x^{15})\) into the final expression.
• Check for like terms if applicable in longer polynomials.
• Simplification ensures faster and more accurate calculations in broader algebraic contexts.

The end result, \(-32x^{15}\), demonstrates the power of these rules in transforming and streamlining equations.