Question

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

Short answer

Question: Rewrite the expression \((-n)^{a+b}\) as a product or a quotient using the rules of exponents. Answer: $$(-n)^{a+b} = (-n)^a(-n)^b.$$

Step-by-step solution

Identify the rules of exponents that apply to the given expression

We are given the expression \((-n)^{a+b}\). First, note that the base is \(-n\) and the exponent is \(a+b\). The rule of exponents that applies to a sum of exponents is \((a^m)(a^n) = a^{m+n}\). In our case, our base is \(-n\) and our exponents are \(a\) and \(b\).

Apply the rule of exponents

Using the rule \((a^m)(a^n) = a^{m+n}\), we can rewrite our expression as: \((-n)^{a+b} = (-n)^a(-n)^b\). That is, we separate the exponent \(a+b\) into two expressions, each with its own power of \(-n\).

Rewrite the expression as a product or a quotient

Our final answer is a product of two expressions, as required: $$(-n)^{a+b} = (-n)^a(-n)^b$$

Key concepts

Rules of Exponents

Understanding the rules of exponents is crucial when working with algebraic expressions involving powers. These rules help simplify expressions and make calculations more manageable.

• Product of Powers: When multiplying two powers that have the same base, you can add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising a power to another power, multiply the exponents. For example, \((a^m)^n = a^{m\times n}\).
• Negative Exponents: An expression with a negative exponent represents a reciprocal. For example, \(a^{-m} = \frac{1}{a^m}\).

These rules enable us to transform complex expressions into simpler forms, which is particularly helpful when solving equations or evaluating algebraic expressions.

Algebraic Expressions

Algebraic expressions include numbers and variables combined using arithmetic operations such as addition, subtraction, multiplication, and division. The expressions can vary from simple to complex forms. Here are some elements found in algebraic expressions:

• Variables: Symbols that represent unknown values, commonly seen as \(x\), \(y\), or \(n\).
• Constants: Numbers with fixed values in the expression, such as 3, -5, or 7.
• Coefficients: Numbers that multiply variables, like 3 in 3\(x\).
• Exponents: Indicate how many times to multiply the base by itself, such as \(n^2\) meaning \(n \times n\).

By mastering the manipulation of these components, solving for unknowns and simplifying expressions becomes more intuitive. Understanding how to work with grouped terms and apply exponent rules brings clarity to even the most daunting algebraic expressions.

Multiplying Exponents

When multiplying terms with exponents, particularly those with the same base, it's important to apply the right rules to simplify the process. In the given exercise, the expression \((-n)^{a+b}\) is separated into two parts, \((-n)^a\) and \((-n)^b\), by applying the exponent rule of a sum.
Here's how it works:

• Identify the base, which remains constant during multiplication.
• Combine the exponents if the operation permits this simplification, particularly when the exponents are added as a single term such as \(a + b\).

To multiply powers effectively:

• Ensure the bases are identical, which allows the addition of exponents (when necessary).
• Apply appropriate exponent rules, like the product of powers, to achieve a more compact form.

This step-by-step approach simplifies expressions like \((-n)^{a+b}\) into more manageable pieces, facilitating easier computation or further algebraic manipulation.