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Chapter 5: Problem 129

Multiply. Assume \(n\) is a natural number. $$ a^{2 n}\left(a^{n}+a^{2 n}\right) $$

Short Answer

Expert verified
The simplified expression is \(a^{3n} + a^{4n}\).

Step by step solution

01

Distribute the terms

We'll start by distributing the term \(a^{2n}\) across the expression inside the parentheses: \[a^{2n} \times (a^n + a^{2n}) = a^{2n} \times a^n + a^{2n} \times a^{2n}\]
02

Simplify each part using the laws of exponents

Now, simplify each part of the expression using the property of exponents that states \(a^m \times a^n = a^{m+n}\).For the first term: \[a^{2n} \times a^n = a^{2n+n} = a^{3n}\]For the second term: \[a^{2n} \times a^{2n} = a^{2n+2n} = a^{4n}\]
03

Combine the simplified terms

Combine the results from the previous step to form the simplified expression:\[a^{3n} + a^{4n}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Numbers
Natural numbers are positive integers starting from 1, going up to infinity. They are the simplest kind of numbers we use in everyday life. These include numbers like 1, 2, 3, and so on. They do not include zero or any negative numbers, fractions, or decimals.
  • Natural numbers are often denoted by the symbol \(\mathbb{N}\).
  • They are always whole numbers, meaning they have no fractional or decimal part.
  • In the context of our exercise, a natural number would be values of \(n\) like 1, 2, 3, etc.
Understanding that \(n\) is a natural number helps us know that when exponentiating, the exponents will always be non-negative integers, which simplifies calculations considerably.
Laws of Exponents
Using the laws of exponents makes calculations much easier and allows us to simplify expressions efficiently. The main rule used in this exercise is the product of powers property, which states:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Exponent of an exponent can be multiplied: \((a^m)^n = a^{m\cdot n}\)
In the exercise, these rules help simplify multiplication of like bases.
Let's break down the use of the laws in the solution:
  • First term simplification: For \(a^{2n} \times a^n\), using the product of powers property, we add exponents: \(a^{2n+n} = a^{3n}\).
  • Second term simplification: For \(a^{2n} \times a^{2n}\), we also add exponents: \(a^{2n+2n} = a^{4n}\).
Using these laws makes the process of simplifying exponentials straightforward, making complex algebraic expressions much simpler to handle.
Distributive Property
The distributive property is a fundamental property of algebra used to simplify expressions. It states that multiplying a single term by a sum inside a parenthesis is the same as doing the multiplication individually to each sum's term.
This is represented mathematically as:
  • \(a(b + c) = ab + ac\)
In the current exercise, we see this at play where \(a^{2n}(a^n + a^{2n})\) uses the distributive property to expand:
  • First, distribute \(a^{2n}\) to \(a^n\), which forms \(a^{2n} \times a^n\).
  • Then, distribute \(a^{2n}\) to \(a^{2n}\), forming \(a^{2n} \times a^{2n}\).
After using the distributive property, we can then use the laws of exponents to further simplify the expression, which helps us get to the final answer efficiently and without mistakes.

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