Problem 45 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (-3 c d)^{3}\left(-d^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(27c^3d^5\).

Step 1: Simplify the cubic power

First, apply the power of a product rule to \((-3 c d)^{3}\), this gives \((-3)^3 * c^3 * d^3\). This simplifies to \(-27 c^3 d^3\).

Step 2: Multiply with \(-d^{2}\)

Next, multiply the previous result, which should be \(-27c^3d^3\), with \(-d^{2}\). This results in \(27c^3d^3d^2\).

Step 3: Simplify the Product of Powers

Finally, simplify the product of powers: \(27c^3d^3d^2\), by adding the powers of \(d\). This results in the final solution \(27c^3d^5\).\(-\).

Key Concepts

Power of a Product Rule

Imagine you're tasked with raising a group of factors to a power—this is where the power of a product rule comes into play. Quite straightforward, it states that when you have a product (which means items being multiplied together) inside parentheses, followed by an exponent, you can distribute that exponent to each factor within the parentheses. This is like giving an invitation to each member of a team to an event: every single one gets one, no one is left out.

Let's work through an example using numbers and variables: \( (-3cd)^3 \). Here, \( -3 \) is the numerical factor, and \( c \) and \( d \) are the variable factors. The power of a product rule tells us to raise each factor to the third power, individually: \( (-3)^3 \) and \( c^3 \) and \( d^3 \). After you calculate \( (-3)^3 = -27 \) and put it all back together, you get \( -27c^3d^3 \)—a neat, simplified expression.

Product of Powers

When our algebraic journey leads us to multiply powers with the same base, we can use the product of powers rule, a kind of shortcut in math. For instance, if you travel from one city to another and then continue to a third city, instead of breaking down the trip into two separate parts, you can just add the travel distances together for the total distance covered.

Mathematically speaking, when you come across an expression such as \( d^3*d^2 \)—where both terms have \( d \) as the base—you simply add the exponents: \( 3 + 2 \). It means you are essentially multiplying \( d \) by itself three times, and then by itself another two times, which amounts to five times in total. That is \( d^5 \), a much simpler way of showing the same product. This rule keeps your expressions compact and your calculations quick.

Simplifying Cubic Power

Dealing with cubic powers, that is to say, exponents of three, can sound daunting, but it's just an elevated form of multiplication. Simplifying cubic power is like unpacking a cube-shaped gift—you reveal the layers one by one until you reach the core.

Think about \( (-3cd)^3 \), which we encountered earlier. To simplify this cubic power using the steps outlined before, you raise each component inside the parentheses to the third power. It may help to visualize this as three copies of \( -3cd \) being multiplied together. After you've applied the power to each factor—to \( -3 \) resulting in \( -27 \), and to \( c \) and \( d \) resulting in \( c^3 \) and \( d^3 \)—your 'cube' is fully unwrapped, revealing a simpler expression: \( -27c^3d^3 \).

Remembering the Orders

Keep in mind, this process works because multiplication is commutative; you can multiply numbers in any order and still get the same answer. This flexibility allows you to group and power-up expressions confidently, simplifying even the most complex-looking cubic powers into digestible pieces.