Problem 83 - Algebra 1

Question

Simplify the variable expression. $$\left(\frac{1}{8}\right)(-4)(-x)(-x)$$

Step-by-Step Solution

Answer

-0.5x^2

Step 1: Evaluate the numerical expression

Evaluate the numerical part of the expression first. This means multiplying \(\frac{1}{8}\) by -4 which equals to -0.5.

Step 2: Multiply by Negative Variables

Next, multiply -0.5 by -x, which equals to positive 0.5x because the product of two negatives is a positive. Repeat the same process once again, 0.5x multiplied by -x equals to -0.5x^2 due to the fact that any term multiplied by the same variable gets exponentiated.

Step 3: Simplify Further if Possible

In this case, the expression cannot be simplified any further. Hence, -0.5x^2 is the simplified form of the given variable expression.

Key Concepts

Simplify Expressions

Simplifying expressions means making a math problem look easier to understand or shorter without changing its value.
When you have an algebraic expression like \(\left(\frac{1}{8}\right)(-4)(-x)(-x)\), your goal is to combine like terms and factors, and remove any unnecessary elements.
Here’s how it works:

• First, look for similar numbers or terms you can multiply or divide.
• Focus on any constants (just numbers) first.
• Then, combine terms that contain the same variables by following the rules of algebra.

In our example, the first step was to multiply \(\frac{1}{8}\) by \(-4\) resulting in \(-0.5\).
We needed to simplify further by working with the variable \(-x\) and using the rules of exponents.

Negative Numbers

Negative numbers can sometimes seem a bit tricky, but they follow clear rules. When dealing with them:

• Remember, two negatives equal a positive when multiplied or divided.
• When adding or subtracting, think of negative numbers as "going in the opposite direction".

In our example:- First, combine \(\frac{1}{8}\) with \(-4\), taking care of the negative sign to get \(-0.5\).- Next, multiplying \(-0.5\) by \(-x\) gives a positive \(0.5x\).
Remember that a negative and a negative give a positive when multiplied.
The process repeats when \(0.5x\) is then multiplied by another \(-x\), resulting in a negative \(-0.5x^2\).
It's crucial to keep track of these negative signs when simplifying expressions.

Variables

Variables are symbols in math, usually letters, that stand for unknown values. They allow you to write expressions that can work with many numbers.
In algebra, variables follow certain rules:

• They can be added, subtracted, multiplied, or divided just like numbers.
• When you multiply them, they may change into powers or become part of exponents.

In our task:- We had the variable \(-x\). Multiplying \(-x\) by itself is like saying \(-x\times -x=x^2\).
This is because multiplying a negative by a negative produces a positive, and variables when multiplied are written with exponents.

Exponents

Exponents are a way to show how many times a number or variable is multiplied by itself.
They’re written as a small number to the top right of a base number or variable, like \(x^2\). Here's what you need to know:

• An exponent of 2 means you multiply the base by itself.
• Exponents make expressions look cleaner and are essential for simplifying.

In our example, you end up with \(-0.5x^2\).- The \(x^2\) comes from multiplying \(-x\) by \(-x\), showing the variable was used twice.
Keep in mind that exponents help group repeated multiplication, making expressions more manageable.
Understanding and using exponents well will simplify algebraic problems a lot.