Problem 80 - Elementary Algebra

Question

For the following problems, write the expressions using exponential notation. \(x\) cubed plus 2 times \((y-x)\) to the seventh.

Step-by-Step Solution

Answer

Question: Rewrite the expression "x cubed plus 2 times (y-x) to the seventh" using exponential notation. Answer: \(x^3 + 2(y-x)^7\)

Step 1: Identify the first term

The first term in the expression is "x cubed," which can be written using exponential notation as \(x^3\).

Step 2: Identify the second term

The second term in the expression is "2 times (y-x) to the seventh." The "to the seventh" part indicates an exponent of 7. So, we can write this term using exponential notation as \(2(y-x)^7\).

Step 3: Combine the terms

Now that both terms are in exponential notation, we can combine them to write the full expression. The expression is the sum of the two terms, so the final answer is \(x^3 + 2(y-x)^7\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the building blocks of algebra and help us describe mathematical situations in concise ways. In algebraic expressions, each component is essential:

• **Variables**: Symbols that stand for unknown values, like \(x\) and \(y\) in our exercise.
• **Constants**: Specific numbers used in the expression, such as the number 2 in \(2(y-x)^7\).
• **Operations**: Mathematical actions like addition, subtraction, multiplication, or division.

In our given problem, the expression \(x^3 + 2(y-x)^7\) consists of two terms:

• The first term is \(x^3\), which involves the variable \(x\).
• The second term is \(2(y-x)^7\), which includes a constant (2), a subtraction operation \((y-x)\), and an exponent.

Understanding how these elements function together allows us to manipulate expressions to simplify, solve, or rewrite them in different forms.

Exponents

Exponents are a shorthand way of expressing repeated multiplication. They show how many times a number, called the "base," is multiplied by itself. In our exercise, we encounter the exponent on \(x\) with \(x^3\) and on \(y-x\) with \((y-x)^7\).

• **Base and Exponent**: In \(x^3\), \(x\) is the base, and 3 is the exponent, meaning \(x\) is multiplied by itself three times: \(x \cdot x \cdot x\).
• **Order of Operations**: When dealing with exponents, it's essential to respect the "order of operations" (PEMDAS/BODMAS), so calculations with exponents come before multiplication or addition in expressions.

Exponents simplify expressions by reducing lengthy multiplication sequences into compact forms. This allows for easier manipulation and analysis of more complex expressions. For instance, the term \((y-x)^7\) shows that the expression \((y-x)\) is multiplied by itself seven times, which can be quite cumbersome to write otherwise.

Polynomials

Polynomials are a type of algebraic expression that can have multiple terms. These terms can typically include variables raised to whole number exponents and coefficients.

• **Structure of Polynomials**: A polynomial consists of sum of terms, like \(x^3\) and \(2(y-x)^7\), each having their variables and exponents.
• **Degree of Polynomial**: The degree is determined by the highest exponent in the expression. For instance, in our example, the degree of the whole polynomial would be 7 due to the term \((y-x)^7\).

Polynomials are versatile and can describe a wide range of real-world phenomena. They can be added, subtracted, multiplied, and even factored, opening doors to various solution strategies and simplifications. In the given expression \(x^3 + 2(y-x)^7\), the combination of terms makes it a classic algebraic polynomial, providing an excellent example of how different algebraic components come together in mathematical modeling.