Question

Multiply and simplify. $$\left(2 a^{x} b c^{2}\right)\left(3 a^{2} b c^{x}\right)$$

Short answer

The simplified result is \(6a^{x+2}b^2c^{x+2}\).

Step-by-step solution

Multiplication of coefficients

First, multiply the numerical coefficients of the terms. In this case, multiply 2 by 3.

Multiplication of like bases with exponents

For the variables with the same base, add the exponents according to the laws of exponents. Multiply the terms with bases of 'a', 'b', and 'c' separately.

Write the simplified expression

Combine the results from step 1 and 2 to form the simplified expression.

Key concepts

Laws of Exponents

Understanding the laws of exponents is crucial when you are multiplying expressions that contain powers. These laws are like shortcuts that help you manipulate exponents quickly and accurately. For example, when you are multiplying terms with the same base, you simply add their exponents. This is based on the power rule which states that for any nonzero number 'a' and any integers 'x' and 'y', the rule is:
\( a^{x} \times a^{y} = a^{x+y} \).
There are more rules like the product rule, the quotient rule, and the power of a power rule, but in the context of multiplying expressions, adding the exponents of like bases is often the focus. It's also important to remember that any number, except zero, to the power of zero is one. Hence, knowing these laws helps to simplify the given algebraic multiplication into a form that is much more manageable.

Simplifying Expressions

When we talk about simplifying expressions, we're essentially discussing how to make an algebraic expression more compact and easier to understand. To simplify an expression, we combine like terms, which in terms of exponents means adding the exponents of the same bases, as discussed in the laws of exponents. We also perform operations with coefficients. Simplifying helps in reducing any complexity and it makes the expressions more straightforward for further operations, like solving equations.

During simplification, maintaining the correct order of operations is also vital—often encapsulated in the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In the context of our exercise, simplification involves multiplying numbers and adding exponents step by step to ensure each operation is handled properly and the simplest form is achieved.

Algebraic Multiplication

The term algebraic multiplication may sound daunting, but it follows the same principles we use for multiplying numbers, just with variables included. When multiplying algebraic expressions, we multiply coefficients together and variables with the same base together, while adding their exponents as per the laws of exponents.
For instance, in our example:
\( (2 a^{x} b c^{2}) \times (3 a^{2} b c^{x}) \),
the coefficients 2 and 3 are multiplied to give 6. For the variables, since 'a', 'b', and 'c' are present in both terms, their exponents are added together following the law: \( a^{x} \times a^{2} = a^{x+2} \), and similarly for 'b' and 'c'. The concept is very much like stacking blocks, with every similar block adding to the height—the height, in this case, being the exponent of the variable. This process is a foundational skill in algebra and is indispensable in higher-level mathematics, physics, and engineering.