Question

Divide and simplify. $$36 d^{5}-6 d^{2} \text { by } 3 d$$

Short answer

The simplified expression after dividing \(36d^5 - 6d^2\) by \(3d\) is \(12d^4 - 2d\).

Step-by-step solution

Identify the Term to Divide by

The expression that needs to be divided is \(36d^5 - 6d^2\). The term to divide by is \(3d\). Ensure that each term in the expression is divided separately by \(3d\).

Divide Each Term Individually

Divide the first term \(36d^5\) by \(3d\) to get \(12d^4\). Then divide the second term \(6d^2\) by \(3d\) to get \(2d\). Use the properties of exponents to simplify the division of terms with the same base.

Write the Simplified Expression

Combine the simplified terms to obtain the final expression. After division, the resulting expression is \(12d^4 - 2d\). This is the simplified form of the given expression after dividing by \(3d\).

Key concepts

Simplifying Expressions

The process of simplifying expressions is akin to tidying up a cluttered room, making it more organized and easier to understand. When we simplify an expression, we streamline it by combining like terms, factoring, and reducing fractions to their lowest terms.

For instance, when simplifying the algebraic expression
\[\begin{equation}36d^5 - 6d^2\br\br\br\br\end{equation}\]
the goal is to divide each term by
\[\begin{equation}3d\br\br\br\br\end{equation}\]
effectively reducing clutter. Just as you might organize books by size on a shelf, you organize terms based on their algebraic components. This act of simplification helps in clearly understanding the structure of the expression and makes subsequent algebraic operations more straightforward.

Properties of Exponents

In Algebra, understanding the properties of exponents is like knowing the rules of a board game—it allows you to play the game effectively and accurately. Exponents communicate how many times to multiply a base number by itself.

To simplify divisions involving exponents with the same base, you can subtract the exponent in the denominator from the exponent in the numerator. For example, given the term
\[\begin{equation}36d^5\end{equation}\]
when dividing by
\[\begin{equation}3d\br\br\br\br\end{equation}\]
we apply the properties of exponents by subtracting the exponent on the 'd' in the denominator from the exponent on the 'd' in the numerator, which results in
\[\begin{equation}36d^ {5-1}=36d^4\br\br\br\br\end{equation}\]
This application simplifies the computation and gives clarity to the process of simplifying algebraic expressions. Always remember, when the base remains the same, exponents can be directly manipulated to simplify the division.

Algebraic Division

Algebraic division, where terms consist of numbers and variables, follows the same principles as numerical division. Think of it as sharing a pie evenly—each slice represents a portion of the whole.

When dividing terms such as
\[\begin{equation}36d^5\end{equation}\]
by
\[\begin{equation}3d\br\br\br\br\end{equation}\]
we distribute the division to each term in the expression. First, divide the coefficients (the numerical parts):
\[\begin{equation}36 \fontfamily{phv}\fontsize{12pt}{16pt}\text{/} 3 = 12\br\br\br\br\end{equation}\]
Next, divide the variables using the properties of exponents, treating the variables as you did the coefficients. This step-by-step method ensures a clear and systematic approach to dividing polynomials, much like following a recipe to ensure each ingredient is properly added to the dish.