Problem 24 - Elementary Algebra

Question

Evaluate \(-2 m(m-3)^{2}\) if \(m=-4\).

Step-by-Step Solution

Answer

Question: Evaluate the expression \(-2m(m-3)^{2}\) for \(m = -4\). Answer: 392

Step 1: Substitute the given value of m in the expression

We are given the expression \(-2 m(m-3)^{2}\) and we are asked to evaluate it for \(m = -4\). Let's substitute the given value of \(m\) into the expression: \[-2(-4)((-4)-3)^2\]

Step 2: Simplify inside the parentheses

Now we need to evaluate the expression inside the parentheses: \[-4 - 3 = -7\] The expression becomes: \[-2(-4)(-7)^2\]

Step 3: Evaluate the exponent

Next, we must evaluate the exponent \((-7)^2\). Remember that any number raised to the power of 2 is equal to that number multiplied by itself: \[(-7)^2 = (-7)(-7) = 49\] Now the expression becomes: \[-2(-4)(49)\]

Step 4: Multiply the numbers

Finally, we need to multiply the numbers in the expression: \[-2 \times (-4) \times 49 = 8 \times 49 = 392\] So, the evaluated expression for \(m = -4\) is 392.

Key Concepts

Substituting Variables

Understanding how to substitute variables is crucial in algebra because it allows you to replace abstract symbols with concrete numbers. This is often the first step when solving equations or evaluating expressions. As in our example exercise, we're asked to evaluate -2 m(m-3)^{2} when m=-4.

The substitution process involves carefully replacing each instance of the variable with the given number. In our case, wherever we see an m, we insert -4, respecting the operations around it. It's like giving the variable m a 'mask' to wear, which in this scene is -4. This step is essential because if done incorrectly, it could potentially change the entire outcome of the expression.

Remember to always keep track of your signs when substituting. If a negative number is being substituted, like -4, and it's within parentheses or being multiplied, be mindful of the resulting sign after substitution.

Simplifying Expressions

After variables are substituted, simplifying the expression is next. Simplification cleans up the equation to make it easier to understand and solve. It's much like organizing a messy room so you can navigate it more easily. The aim is to reduce the expression to as few terms as possible while following the order of operations—parentheses, exponents, multiplication and division, then addition and subtraction (PEMDAS).

As illustrated in our example, once the variable was substituted, the expression within the parentheses had to be resolved first. Simplifying is all about being systematic. When you see something like (-4)-3, calculate what's inside the parentheses to get -7. This may seem mundane, but it is where the beauty of algebra shines—turning something complicated into something simple.

After the parentheses are taken care of, look out for any exponents and deal with them next. Then, move on to multiplication or division, and finally, manage any remaining additions or subtractions. This process helps in avoiding any miscalculations and arriving at the correct and simplified result.

Exponents and Powers

Exponents and powers represent repeated multiplication. For instance, the square of a number, represented as x^2, simply means x multiplied by itself. In our example, we deal with (-7)^2, which follows the rule that a negative number squared results in a positive product—this is a crucial concept to understand. Always remember that an even exponent will turn a negative number into a positive.

To evaluate an exponent, like (-7)^2 = 49, you multiply -7 by itself, resulting in 49. Exponential terms need to be calculated before moving on to other operations, following our PEMDAS rule. By mastering exponents, you'll find that complex expressions become far more approachable. It's like having a secret weapon in your math arsenal that simplifies your numbers swiftly and efficiently.