Problem 67 - Elementary and Intermediate Algebra

Question

Perform each division. $$ \frac{45 m^{10}}{9 m^{5}} $$

Step-by-Step Solution

Answer

The result is \( 5m^5 \).

Step 1: Simplify the Coefficients

Start by dividing the numerical coefficients of the given terms. In the expression \( \frac{45}{9} \), divide 45 by 9. This simplifies to 5.

Step 2: Apply the Quotient Rule for Exponents

Next, use the quotient rule for exponents which states that \( \frac{a^m}{a^n} = a^{m-n} \) when dividing like bases. Apply this to the variable part \( \frac{m^{10}}{m^{5}} \), which simplifies to \( m^{10-5} = m^5 \).

Step 3: Combine Results

Combine the simplified coefficient and the simplified variable terms. The expression becomes \( 5 \cdot m^5 \), which simplifies to \( 5m^5 \).

Key Concepts

Quotient Rule for Exponents

When dealing with exponents in algebra, one of the essential rules to remember is the quotient rule for exponents. This rule helps us simplify expressions that include division of terms with the same base.

• The quotient rule states that whenever you divide powers with the same base, you subtract the exponents. In mathematical terms, it is expressed as \( \frac{a^m}{a^n} = a^{m-n} \).
• This rule simplifies complex expressions, making them easier to handle.

Applying this rule helps reduce computational efforts while solving algebraic equations, as demonstrated in the initial exercise. By applying \( \frac{m^{10}}{m^{5}} \), the exponents are subtracted resulting in \( m^{10-5} = m^5 \).
This method is incredibly useful when working through polynomial divisions because it keeps calculations simplified and manageable.

Simplifying Expressions

Simplifying expressions is a crucial step in algebra that allows us to reduce complex equations to their simplest form. This often involves breaking down the expression into its core components.

• The aim is to make the expression as simple as possible, often by eliminating excess terms and simplifying coefficients and variables.
• For numerical coefficients, simplify them by dividing to find their greatest common factor. For instance, \( \frac{45}{9} = 5 \).
• After simplifying coefficients, focus on the variables using applicable mathematical rules, such as the quotient rule for the exponents.

By following these steps in the expression \( \frac{45m^{10}}{9m^{5}} \), we simplified it to a much easier form: \( 5m^5 \). This helps in further algebraic operations and gives a clearer understanding of the underlying relationships between the variables.

Polynomial Division

Polynomial division shares similarities with arithmetical long division but is applied to polynomials rather than numbers. It allows us to divide one polynomial by another, reducing complexity and revealing more about the polynomial's structure.

• This technique is essential when you want to simplify a polynomial expression or determine a polynomial’s factors.
• When dividing polynomials, you usually start by dividing the leading terms and work your way through the expression.
• Although the example discussed here involved dividing a simple polynomial by a monomial, the approach remains similar, involving simplifying every term.
• It encompasses using various mathematical rules and steps, including the quotient rule for exponents discussed earlier as it often comes into play with similar bases.

By understanding polynomial division, you can simplify and solve complex polynomial equations, allowing for clearer insights into their behavior and characteristics. As seen, we simplified the polynomial \( \frac{45m^{10}}{9m^{5}} \) to \( 5m^5 \) using systematic division and simplification techniques.