What is the product 3a 2b 4 8ab 3 – What is the product of 3a2b48ab3? This mathematical expression may seem daunting at first glance, but by breaking it down into smaller steps, we can unravel its secrets and uncover its underlying mathematical principles.

The expression can be simplified by following the order of operations: first, multiply the coefficients (3 and 4), then multiply the variables (a and b), and finally, add the exponents (2 and 3). This gives us the final product of 96a3b5.

Product Definition: What Is The Product 3a 2b 4 8ab 3

The given mathematical expression, 3a ²b ⁴8ab ³, represents the product of two terms, (3a ²b ⁴) and (8ab ³).

Breaking down the first term, (3a ²b ⁴), we have the coefficient 3, followed by the variable ‘a’ raised to the power of 2, and the variable ‘b’ raised to the power of 4.

Similarly, the second term, (8ab ³), consists of the coefficient 8, the variable ‘a’ raised to the power of 1, and the variable ‘b’ raised to the power of 3.

To find the product of these two terms, we multiply the coefficients, the variables with the same base are multiplied, and the exponents of those variables are added.

Numerical Evaluation

Substituting the given values, a = 2 and b = 3, into the expression, we get:

3(2 ²)3 ⁴8(2)3 ³= 3(4)81(16)27 = 3243227

Therefore, the numerical value of the expression is 3243227.

Mathematical Properties

The expression exhibits several mathematical properties:

• Associative Property of Multiplication:The order of multiplication of the terms does not affect the result. (3a ²b ⁴)8ab ³= 8ab ³(3a ²b ⁴)
• Commutative Property of Multiplication:The order of the factors can be changed without altering the product. 3a ²b ⁴8ab ³= 8ab ³3a ²b ⁴
• Distributive Property of Multiplication over Addition:The expression can be expanded by distributing the coefficients over the variables. 3a ²b ⁴8ab ³= 24a ³b ⁷+ 24a ²b ⁴

Applications and Examples

The product of algebraic terms has applications in various fields, including:

• Geometry:Calculating the area or volume of shapes with algebraic dimensions.
• Physics:Describing physical quantities and relationships, such as force, energy, and momentum.
• Engineering:Designing and analyzing structures and systems.

Historical Context

The concept of multiplying algebraic terms has its roots in ancient mathematics.

In the 9th century, the Persian mathematician Muhammad ibn Musa al-Khwarizmi introduced the concept of coefficients and variables, paving the way for the development of algebraic expressions.

Over the centuries, mathematicians like René Descartes and Leonhard Euler further developed the rules and properties of algebraic operations, including multiplication.

Expert Answers

What is the first step in simplifying the expression?

Multiply the coefficients (3 and 4).

What is the final product of the expression?

96a3b5

In what fields is this expression commonly used?

Algebra and other mathematical fields