Question

Reduce to lowest terms. Write your answers without negative exponents. Do some algebraic fractions by calculator. $$\frac{12 m^{2} n}{15 m n^{2}}$$

Short answer

\frac{4m}{5n}

Step-by-step solution

Factor Numerator and Denominator

First, factor the coefficients (numbers) and the variables in both the numerator and the denominator. The numerator is 12m^2n, which we can keep as is, since 12 is already in its prime factor form, and so are the variables. The denominator is 15mn^2. The number 15 can be factored into its prime components 3 and 5.

Reduce Common Factors

Next, reduce any common factors that appear in both the numerator and the denominator. Here, both 12 and 15 share a common factor of 3. Also, the variables 'm' and 'n' both appear in the numerator and denominator.

Cancel Out Common Terms

Now, cancel out the common terms. The factor of 3 in 12 and 15 will cancel out, reducing them to 4 and 5 respectively. The variable 'm' appears to the power of 1 in the denominator and the power of 2 in the numerator, so we can subtract the exponent in the denominator from the numerator to get m^(2-1). Similarly for 'n', we subtract the power in the numerator from the power in the denominator to get n^(2-1).

Write the Simplified Fraction

Write down the simplified fraction without any negative exponents. After canceling out the common terms, we are left with 4m in the numerator and 5n in the denominator.

Key concepts

Factoring Expressions

Factoring expressions is a key concept in simplifying algebraic fractions. It involves breaking down numbers and variables into their simplest form, typically their prime factors. In the given exercise, factoring helps identify common factors in the numerator and denominator that can be reduced. For coefficients, this means expressing them as a product of prime numbers. For example, the number 12 can be factored into 2 × 2 × 3, which are its prime components. Variables are treated similarly but are broken down based on their exponents. For instance, the variable term with an exponent, such as \(m^2\), indicates that \(m\) appears twice as a factor.
By factoring, we systematically prepare the expression for simplification. This process lays the foundation for the next steps, which involve reducing and eliminating common factors to achieve the simplest form of the algebraic fraction.

Reducing Common Factors

Once we've factored expressions, the next vital step is to reduce common factors. Reducing common factors simplifies the algebraic fraction, by dividing both the numerator and denominator by the same non-zero number or algebraic terms. In the exercise we're discussing, we identified the common factor of 3 in both 12 and 15. Reducing this common factor simplifies the coefficients from 12/15 to 4/5.

Identify and Reduce

It is essential to identify all common factors between the numerator and denominator. In algebraic terms, this includes paying attention to variables that appear in both and reducing them to their lowest powers by subtracting exponents. For example, if a variable such as \(m\) appears as \(m^2\) in the numerator and \(m\) in the denominator, we simplify by subtracting the lower exponent from the higher exponent, leaving us with \(m^{2-1} = m\). By doing so, we're applying the laws of exponents to reduce the fraction step by step.

Cancelling Algebraic Terms

The final crucial step in simplifying an algebraic fraction is cancelling algebraic terms. This step goes hand in hand with reducing common factors, and it involves eliminating the common terms from the numerator and denominator. After reducing common factors, you'll often find terms that are exactly the same on the top and bottom of the fraction.

How to Cancel

To cancel terms, simply divide them out, remembering that any term divided by itself equals one. In our exercise example, after reducing the common coefficient 3, we were left with a remaining variable \(m\) in the numerator and \(n\) in the denominator. Since there's no \(m\) in the denominator and no \(n\) in the numerator left to pair with those in opposite locations, no further cancellation can occur. Thus, we've reached the simplest form: \(\frac{4m}{5n}\). Canceling correctly ensures that we don't lose any information about the variables and their relationships, which is particularly important when solving equations or working with expressions in context.