Problem 59 - Algebra 2

Question

Simplify. Assume that no variable equals \(0 .\) $$ \left(-4 d^{2}\right)^{3} $$

Step-by-Step Solution

Answer

\(-64d^6\)

Step 1: Understand the Problem

The problem asks us to simplify the expression \( \left(-4d^2\right)^3 \). This involves raising the expression inside the parentheses to the power of 3. Our goal is to calculate the result by applying the properties of exponents.

Step 2: Apply the Power to a Product Rule

We can use the rule \( (ab)^n = a^n b^n \) to separate the constant and the variable.In this case, \( \left(-4d^2\right)^3 = (-4)^3 \cdot (d^2)^3 \).

Step 3: Calculate \((-4)^3\)

Calculate the power of the constant term:\[ (-4)^3 = (-4) imes (-4) imes (-4) = -64 \].

Step 4: Calculate \((d^2)^3\)

Apply the power to a power rule, which states that \((a^m)^n = a^{m imes n}\):\[ (d^2)^3 = d^{2 imes 3} = d^6 \].

Step 5: Combine the Results

Now that we have calculated \((-4)^3 = -64\) and \((d^2)^3 = d^6\), combine them to simplify the entire expression:\[ (-4d^2)^3 = -64d^6 \].

Key Concepts

Power to a product rule

To simplify expressions where a product is raised to a power, you can use the "Power to a Product Rule." This rule states that \[ (ab)^n = a^n b^n \] Here, you can see that the power is distributed to each factor within the parentheses. This approach helps in breaking down the simplification process. Suppose you have an expression such as \[ (-4d^2)^3 \]By applying this rule, our expression splits into two parts:

• The constant: \[ (-4)^3 \]
• The variable with an exponent: \[ (d^2)^3 \]

This step-by-step separation makes it easier to attack each part of the expression individually. Each factor raised to the power independently, simplifies calculations and keeps the problem manageable.

Power to a power rule

When dealing with expressions that have exponents, you will often use the "Power to a Power Rule." This is incredibly helpful when you have to simplify terms like \[ (a^m)^n \]. The rule states: \[ (a^m)^n = a^{m \cdot n} \] It simplifies the operation as exponents of exponents can be multiplied directly. Let's look at how this applies to our example: consider the part of the expression \[ (d^2)^3 \]. Using the rule, you multiply the exponents (2 and 3) to get: \[ d^{2 \cdot 3} = d^6 \]. This rule ensures that repeated multiplication is simplified efficiently, transforming what could be a cumbersome expression into something much easier to manage. By converting multiple layers of exponents into a single expression, it reduces mistakes and streamlines calculations.

Simplifying expressions

Once you've applied the rules for each component of an expression, the final step is combining the results to achieve a simpler form. This process is all about making the expression as straightforward as possible, easily understandable and workable for future calculations.In the earlier steps, we applied the Power to a Product rule to split the expression and then used the Power to a Power rule to calculate each term's power. Now we bring it all back together. For \[ (-4)^3 = -64 \] and \[ (d^2)^3 = d^6 \], we combine these results: \[ (-4d^2)^3 = -64d^6 \].By understanding and applying these exponent rules, you can efficiently tackle problems requiring simplification of expressions. Simplifying not only tidies up the expression but also prepares it for any further operations, making algebraic expressions elegant and easy to handle.