Problem 148 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. Find the value of \(m\) if \(m=\frac{-3 k-5 t}{k t+6}\) when \(k=4\) and \(t=-2\).

Step-by-Step Solution

Answer

Answer: The value of m is 1.

Step 1: Substitute the given values

First, substitute the given values of \(k=4\) and \(t=-2\) into the expression for \(m\): \(m=\frac{-3(4)-5(-2)}{(4)(-2)+6}\)

Step 2: Perform the calculations inside the brackets

Next, perform the calculations inside the brackets: \(m=\frac{-12+10}{-8+6}\)

Step 3: Simplify the numerator and denominator

Now, simplify the numerator and denominator: \(m=\frac{-2}{-2}\)

Step 4: Divide the numerator by the denominator

Finally, divide the numerator by the denominator to find the value of \(m\): \(m=1\) So, the value of \(m\) is 1.

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. For example, if you have the expression \(a^n\), then "\(a\)" is the base and "\(n\)" is the exponent. This expression is equal to \(a\times a\times \ldots \times a\), \(n\) times. Exponents allow us to express large numbers succinctly.

In the context of algebraic expressions, it is important to understand how to handle exponents, especially when simplifying expressions. You may encounter negative exponents, which indicate the reciprocal of the base raised to the corresponding positive power. For example, \(a^{-n} = \frac{1}{a^n}\). This property helps in rewriting expressions so that only positive exponents appear. When simplifying expressions, make sure to apply exponent rules consistently to achieve the correct and simplest form.

Simplification

Simplification in mathematics refers to the process of reducing an expression into its simplest or most understandable form. In algebraic expressions, simplification involves combining like terms, applying operations, and reducing fractions to their simplest form.

For the given expression \(m=\frac{-3 k-5 t}{k t+6}\), simplification is crucial after plugging in the given values for \(k\) and \(t\). Following substitution and performing operations inside brackets, you'll often end up with fractions or numbers that can be further simplified by cancelling terms out.

• Simplify by combining like terms where possible.
• Cancel out terms that are common in both the numerator and the denominator.
• Ensure that your final expression only contains positive exponents.

When simplified correctly, expressions are easier to work with, and results can be found more efficiently.

Substitution

Substitution is the process of replacing variables in an expression with their given numerical values. It is a fundamental technique for solving algebraic equations and simplifying expressions when specific variable values are known.

In the exercise, the expression \(m=\frac{-3 k-5 t}{k t+6}\) requires substitution of the provided values of \(k=4\) and \(t=-2\). To substitute:

• Replace each instance of "\(k\)" with 4.
• Replace each instance of "\(t\)" with -2.

This transforms the equation into numbers only, making it easier to calculate and simplify. Verify each substitution step to ensure no mistakes, as they can lead to incorrect final results. Double-checking your substitutions helps avoid missing or misplacing any terms.

Evaluation of Expressions

Evaluation of expressions involves calculating the value of an algebraic expression once all the variables have been substituted with their known values. This step requires careful arithmetic operations and simplification to ensure the final result is correct.

After substituting \(k=4\) and \(t=-2\) into \(m=\frac{-3 k-5 t}{k t+6}\), the next step is to perform the operations:

• First, calculate the values inside the brackets.
• Next, simplify both the numerator and the denominator as individual expressions.
• Finally, divide the simplified numerator by the simplified denominator to solve for \(m\).

In this exercise, the calculations simplified the expression to \(m = \frac{-2}{-2}\), resulting in \(m = 1\). Always perform calculations methodically to avoid errors, and check the logical flow of your work.