Problem 52 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ k(k-11) $$

Step-by-Step Solution

Answer

Answer: The product of k and (k - 11) is k^2 - 11k.

Step 1: Perform the multiplication

To perform the multiplication, apply the distributive property by multiplying the term k with each term inside the parentheses (k - 11): $$ k(k-11) = k \cdot k - k \cdot 11 $$

Step 2: Simplify the multiplication

Now, complete the multiplication for each term: $$ k \cdot k - k \cdot 11 = k^2 - 11k $$

Step 3: Check for like terms

In the resulting expression, there are no like terms to combine. So the final result is: $$ k(k - 11) = k^2 - 11k $$

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by multiple terms inside parentheses. It states that if you have a scenario like \(a(b + c)\), you can distribute the multiplication of \(a\) across both \(b\) and \(c\), resulting in \(ab + ac\).

For our exercise \(k(k - 11)\), applying the distributive property involves multiplying \(k\) by \(k\), and also by \(-11\). This yields \(k \cdot k - k \cdot 11\), an essential step in expanding algebraic expressions. Without the distributive property, combining and simplifying terms would be far more complex, making it a vital tool for understanding algebra.

Like Terms

Like terms are terms in an algebraic expression that have the same variable raised to the same power, but they may have different coefficients. For example, \(3x\) and \(5x\) are like terms, whereas \(3x^2\) and \(5x^3\) are not, because the exponents differ.

In the given solution, \(k^2 - 11k\), there are no like terms. This is because one term is \(k\) squared and the other is \(k\) to the first power. If there had been another term with \(k^2\) or another term with \(k\) (not squared), we would have combined them by adding or subtracting their coefficients as appropriate. Understanding like terms is crucial when simplifying expressions to their most reduced form.

Simplifying Expressions

Simplifying expressions in algebra means reducing them to their simplest form by performing all possible arithmetic operations and combining like terms. It's the process of making the expression as compact and straightforward as possible.

After using the distributive property in the exercise, we carried out multiplication resulting in the expression \(k^2 - 11k\). Since there are no like terms, this expression is already in its simplest form. Simplification makes expressions easier to interpret and often enables a clearer understanding of what the expression represents or how it can be used in equations and inequalities.

Algebraic Multiplication

Algebraic multiplication involves multiplying variables, numbers, and expressions. When multiplying two algebraic terms, you multiply the coefficients (numerical parts) and the variables separately. The power of a variable is determined by the exponent; when you multiply variables with the same base, you add the exponents together.

During the multiplication in our example, \(k \cdot k\) became \(k^2\) because we add the exponents- both of which are implicitly 1, resulting in an exponent of 2 for \(k\). This multiplication is different from arithmetic multiplication because it combines not just numbers but also variables which can represent unknowns or varying quantities.