Problem 23 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression, or explain why the expression cannot be simplified. $$27 x^{3}-26 x^{3}$$

Step-by-Step Solution

Answer

The simplification of the algebraic expression \(27x^{3} - 26x^{3}\) is \(53x^{3}\).

Step 1: Identify the algebraic expressions

The given algebraic expressions are \(27x^{3}\) and \(-26x^{3}\). These expressions have the same base 'x' and the same exponent '3' but different coefficients, which are 27 and -26 respectively.

Step 2: Apply the subtraction operation

Simplify these expressions by subtracting \(-26x^{3}\) from \(27x^{3}\). Just like in simple arithmetic subtraction, subtract the coefficients while the base 'x' and exponent '3' remain the same.

Step 3: Perform the subtraction

Perform the subtraction operation on the coefficients 27 and -26. This operation results in \(27 - (-26) = 27 + 26 = 53\). Therefore, the simplified algebraic expression is \(53x^{3}\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation such as addition, subtraction, multiplication, or division. These expressions create a framework for solving real-world problems and modeling scenarios. In the given exercise, we are dealing with expressions that include variables and constants in a specific format. For example, the expression \(27x^{3}\) consists of:

• A coefficient (27),
• A variable (\(x\)),
• An exponent (3).

These elements are combined to form the complete expression and are foundational in algebra. Always keep in mind that terms in algebraic expressions can often be combined or simplified if they share common variables and exponents.

Coefficients

Coefficients are numerical values that multiply the variables in an algebraic expression. In the expression \(27x^{3}\), the coefficient is 27. They are essential because they determine the magnitude or scale of the term in which they are present. Understanding coefficients allows one to manipulate and simplify expressions effectively.The step-by-step solution to the exercise involves combining coefficients. Since both terms share the same variable and exponent, their coefficients (27 and -26) can be directly subtracted. The new coefficient resulting from this operation will be applied to the variable term, keeping it consistent with the original form.

Exponents

Exponents in algebra serve as a shorthand notation to denote repeated multiplication of a number by itself. The expression \(x^{3}\) represents \(x\times x\times x\). Exponents are vital for representing terms in a more compact and readable form. They also play a crucial role in simplifying expressions by providing the power needed to maintain consistency between similar terms.Because both expressions in the exercise share the same exponent (3), we can focus on subtracting just the coefficients. When adding or subtracting algebraic expressions, ensure that terms have the same base and exponent before performing operations on the coefficients.

Subtraction Operation in Algebra

Subtraction in algebra works in a manner similar to basic arithmetic. Only terms with identical variables and exponents can be subtracted directly. In this exercise, both terms contain \(x^{3}\), which permits the subtraction of coefficients. The expression \(27x^{3} - 26x^{3}\) illustrates this.

• First, identify the coefficients (27 and -26).
• Subtract the second coefficient from the first: \(27 - (-26)\).
• Dealing with negative values in subtraction often leads to adding instead: \(27 + 26 = 53\).

After completing these steps, the result is \(53x^{3}\), which simplifies the initial expression. This negative subtraction operation showcases how algebraic principals simplify broader calculations easily."