Problem 78 - Elementary Algebra

Question

For the following problems, write the expressions using exponential notation. \((y+2)\) cubed.

Step-by-Step Solution

Answer

Question: Rewrite the expression \((y+2)^3\) in exponential notation. Answer: \((y+2)^3\)

Step 1: Identify the base and exponent

In the expression \((y+2)^3\), the base is \((y+2)\) and the exponent is 3.

Step 2: Rewrite the expression in exponential notation

The expression \((y+2)\) cubed can be written in exponential notation as: \((y+2)^3\) By following these steps, we have successfully rewritten the expression in exponential notation as \((y+2)^3\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as add, subtract, multiply, and divide). The purpose of an algebraic expression is to represent a certain quantity or relationship in a concise and symbolic way. For instance, the algebraic expression \(y + 2\) denotes the sum of a variable \(y\) and the number 2. Algebraic expressions are foundational in understanding and solving a variety of mathematical problems, from simple equations to more complex formulations.

In the context of our exercise, \(y + 2\) cubed is an algebraic expression that shows not only a relationship between \(y\) and 2 but also introduces the use of exponents, indicating the power to which the quantity \(y + 2\) is raised.

Exponents and Powers

Exponents and powers are crucial components in mathematics that allow for the representation of repeated multiplication. An exponent, which is placed as a superscript right after a base number or variable, denotes how many times the base is multiplied by itself. For example, in the expression \(x^2\), the number 2 is the exponent, suggesting that \(x\) should be multiplied by itself once (because \(x^1 = x\)), yielding \(x*x\) or \(x^2\). Understanding how to use exponents effectively can simplify complex calculations and is a key concept in algebra and beyond.

Referring to our original problem, \(y+2\) cubed is expressed using exponential notation as \(y+2)^3\), where 3 is the exponent, meaning the previous algebraic expression \(y+2\) is multiplied by itself two more times.

Polynomial Expressions

Polynomial expressions are algebraic expressions consisting of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial is \(3x^2 - 2x + 7\). Polynomials can be classified by the number of terms they have (monomials, binomials, trinomials, etc.) or by their degree, which is the highest exponent of the variable in the expression.

When we raise a binomial like \(y+2\) to a power, such as cubing it to get \(y+2)^3\), the result is a polynomial. This is because the expansion of this expression will involve terms like \(y^3\), \(y^2\), \(y\), and constant numbers, all of which make up a polynomial. Understanding how to work with polynomial expressions is essential for solving a wide range of problems in algebra.