Problem 31 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(7 m\left(\frac{5}{m}\right)\)

Step-by-Step Solution

Answer

The simplified expression is 35.

Step 1: Distribute

Begin by distributing the 7m to the fraction \(\frac{5}{m}\). This gives us \(7m \times \frac{5}{m}\).

Step 2: Multiply the Numerator

Multiply the numerator (7m and 5) to get \(35m\). This results in the expression \(\frac{35m}{m}\).

Step 3: Simplify the Expression

The expression \(\frac{35m}{m}\) simplifies by canceling the common term \(m\) from the numerator and denominator, leaving us with 35.

Key Concepts

Simplifying Fractions

Simplifying fractions is a crucial concept in algebra. It's about reducing fractions to their simplest form so that they are easier to work with. When we have a fraction, it comprises a numerator (the top number) and a denominator (the bottom number). To simplify, you need to look for any common factors that are in both the numerator and denominator.

If you can divide both by a common number or term, you cancel these out. For example, in the fraction \(\frac{35m}{m}\), both the numerator and the denominator have \(m\). Canceling out the \(m\)s leaves us with \(35\).

• Find common factors in the numerator and the denominator.
• Cancel out these factors by dividing both by the greatest common factor.
• The goal is to make the fraction as simple as possible.

Distributive Property

The distributive property is essential when solving expressions in algebra. It allows you to multiply a single term with a group of terms within parentheses. The distributive property states that you multiply the term outside the parentheses by every term inside. So, for an expression like \(7m(\frac{5}{m})\), you multiply \(7m\) by the fraction \(\frac{5}{m}\).

This property ensures that each term within the parentheses is taken into account in a straightforward way:

• The general formula is \(a(b + c) = ab + ac\).
• Breaking it down simplifies computation, especially when variables or fractions are involved.
• Useful for simplifying expressions before simplifying fractions.

Multiplication in Algebra

Multiplication in algebra involves multiplying coefficients and variables just like numbers. When you multiply variables, you look at any coefficients first. If variables are the same, you add their exponents, otherwise, you multiply them as they are.

For expressions like \(7m \times \frac{5}{m}\), first multiply the coefficients \(7 \times 5 = 35\). When multiplying variables, if a variable appears in both the numerator and the denominator, they can often cancel each other out.

• Multiply numerators and denominators separately.
• Cancel out any common terms in multiplication and division if they appear both in numerator and denominator.
• Always simplify the expression by reducing fractional expressions.