Question

Write each expression as a product or a quotient. Assume all variables are positive. $$ (x+1)^{a b+c} $$

Short answer

Question: Rewrite the expression $(x+1)^{ab+c}$ as a product or quotient. Answer: $(x+1)^{ab+c} = (x+1)^{ab} \cdot (x+1)^{c}$

Step-by-step solution

Identify the exponent rule used to divide the exponent

The rule we will be using is: $$ a^{m+n} = a^m \cdot a^n $$ Here, the expression \((x+1)^{ab+c}\) has a sum in the exponent, so we want to split it into two separate terms using the given rule.

Break the complex exponent into simpler parts

Applying the rule to the expression: $$ (x+1)^{ab+c} = (x+1)^{ab} \cdot (x+1)^{c} $$ We can break the complex exponent into two simpler parts: \(ab\) and \(c\). Now, the expression is in the form of a product as instructed.

Finalize the rewritten expression

We have successfully expressed the given expression as a product, so the final expression is: $$ (x+1)^{ab+c} = (x+1)^{ab} \cdot (x+1)^{c} $$

Key concepts

Algebraic Expressions

Algebraic expressions are collections of numbers, variables, and operations that represent a particular quantity or relationship. These expressions form the backbone of algebra and help you translate real-world situations into mathematical language. In our exercise, we focused on an example with algebraic expressions such as \((x+1)^{ab+c}\), where \(x+1\) is a part of the expression and serves as the base in the context of exponentiation.

Here's what you need to know about algebraic expressions:

• Variables: Symbols like \(x\) represent values that can change.
• Constants: Definite numbers like 1, 2, or any specific value that does not change.
• Operations: Include addition, subtraction, multiplication, and division.

Algebraic expressions can be manipulated using various rules and properties. Understanding these fundamental concepts allows you to break down and rearrange expressions, therefore making complex problems easier to solve. Recognizing how an expression like \((x+1)^{ab+c}\) can be rewritten using logical rules is essential in facilitating more straightforward problem-solving.

Exponents

Exponents are a shorthand way to denote repeated multiplication. In our example, we dealt with expressions involving exponents, such as \((x+1)^{ab+c}\). Here the expression indicates that \(x+1\) should be multiplied by itself \(ab+c\) times.

Understanding exponentiation is critical:

• An exponent tells us how many times to use the base in a multiplication.
• The exponent rule \(a^{m+n} = a^m \cdot a^n\) is useful for simplifying expressions with exponents. For example, \((x+1)^{ab+c}\) can be separated using this rule.
• This rule works because of the associative property of multiplication.

When you rewrite the exponent from \((x+1)^{ab+c}\) to \((x+1)^{ab} \cdot (x+1)^c\), it splits the expressions into simpler parts, which can be extremely useful for evaluating or further manipulating algebraic expressions. Mastering these exponent rules helps significantly when solving complex mathematical problems.

Mathematical Notation

Mathematical notation is a system of symbols used to write formulas or equations. Understanding this notation is essential for reading and writing mathematical concepts clearly and accurately.

In this exercise, we used the notation \((x+1)^{ab+c}\) to express an expression with an exponent. Mathematical notation helps:

• Simplify communication of complex ideas through symbols and formulas.
• Standardize communication in the math community globally, ensuring everyone has a consistent understanding of expressions and equations.
• Enhance clarity and precision in mathematical communication.

It's crucial to get comfortable with mathematical notation, because it enables you to work efficiently with algebraic expressions and other formulas beyond this specific example. This proficiency helps you follow mathematical transformations and understand the rules applied throughout various steps of problem-solving, just like how we applied the exponent rule to simplify \((x+1)^{ab+c}\).