Problem 28 - Elementary Algebra

Question

For the following problems, use algebraic notataion. zero minus \(a\) times \(b\)

Step-by-Step Solution

Answer

Question: Write the expression "zero minus a times b" in algebraic notation and simplify. Answer: -ab

Step 1: Write down the given expression in algebraic notation

Given expression is: zero minus \(a\) times \(b\). We write it as \(0 - a \times b\).

Step 2: Multiply \(a\) and \(b\)

When you multiply any two variables, you write them together without any multiplication sign. So, multiply \(a\) and \(b\) will be written as \(ab\).

Step 3: Subtract the product from zero

Now, we need to subtract the result from zero. Since the product is \(ab\), the expression becomes \(0 - ab\).

Step 4: Simplify the expression

Since there are no like terms to combine, the simplified expression is \(-ab\). Therefore, zero minus \(a\) times \(b\) is written in algebraic notation as \(-ab\).

Key Concepts

Multiplication of Variables

In algebra, multiplying variables is straightforward. Variables represent unknown quantities and when we multiply them, we simply write them side by side. For example, if we have variables \(a\) and \(b\), their multiplication is written as \(ab\). No multiplication sign is required, keeping it clean and simple.

• This shows the product of the variables.
• It's a compact way of writing multiplication in algebra.

If you encounter another multiplication, like between \(a\) and \(b\) again, it results in \(ab = ba\), due to the commutative property of multiplication. It's important to remember that the order of multiplication doesn't affect the result. Even though the variables are symbols, they follow the same rules as regular numbers in multiplication.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations, like addition, subtraction, multiplication, and division.

• They can be as simple as a single variable \(x\) or a number like \(3\).
• They can also be more complex, like \(2a + 3b - 4\).

An algebraic expression allows us to write calculations in a shorthand form using variables to represent numbers. This allows us to generalize or describe situations algebraically before plugging in specific values. For instance, "zero minus \(a\) times \(b\)" becomes \(0 - ab\) using algebraic expressions. This way, the essential mathematical operation is clear and concise.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This often means removing unnecessary steps or symbols while keeping the essential information intact. In the case of subtraction from zero, like \(0 - ab\), simplifying it results in \(-ab\).

• This is because subtracting a positive term is equivalent to adding its negative.
• There are no like terms in \(-ab\) since it's already as simple as it can be.

Through simplification, expressions are easier to read and work with. It also prevents errors in more complex calculations, as you are always starting from the simplest form possible. Simplifying is a key practice in algebra to take the clutter out of equations and expressions.