Problem 630 Simplify each expression. $$n ... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 630

Simplify each expression. $$n \cdot n^{4}$$

Short Answer

Expert verified

$n^5$

Step by step solution

Identify the Expressions

Recognize that the expression given is a product of two terms: $n$ and $n^4$.

Apply the Product of Powers Rule

Use the product of powers rule, which states that when multiplying two expressions with the same base, you add the exponents: $a^m \times a^n = a^{m+n}$.

Add the Exponents

Apply the rule: $n \times n^4$ becomes $n^{1+4}$.

Simplify the Exponent

Combine the exponents: $n^{1+4} = n^5$.

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Key Concepts

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

Product of Powers Rule

To simplify the expression, we need to apply the product of powers rule. This rule is essential when dealing with exponents. It states that when multiplying two or more expressions with the same base, you simply add the exponents together. For example, if you have the expression $a^m \times a^n$, then it simplifies to $a^{m+n}$. This rule works because the exponents indicate how many times the base is used in multiplication. By adding them, we combine these repetitions into a single exponentiation.

Algebraic Manipulation

Simplifying expressions often requires algebraic manipulation. This means using rules and properties of algebra to rewrite the expression in a simpler form. In our exercise, we started with $n \times n^4$. By recognizing that both terms have the same base ($n$), we were able to apply the product of powers rule to combine the exponents. This step-by-step process of recognizing patterns and applying algebraic rules makes complex expressions much easier to handle. Remember to always start by identifying the parts of the expression before applying any rules.

Exponents

Exponents, or powers, are a shorthand way to express repeated multiplication of a number by itself. For example, $n^4$ means $n$ is multiplied by itself four times: $n \times n \times n \times n$. When working with exponents, it's important to understand several key rules:

Product of Powers Rule: When multiplying two powers with the same base, add their exponents (e.g., $n^1 \times n^4 = n^{1+4} = n^5$).
Power of a Power Rule: When raising a power to another power, multiply the exponents (e.g., $(a^m)^n = a^{m \times n}$).
Quotient of Powers Rule: When dividing two powers with the same base, subtract the exponents (e.g., $a^m / a^n = a^{m-n}$).

Understanding these rules will help you manipulate and simplify expressions efficiently.

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91影视

Simplify each expression. $$n \cdot n^{4}$$

Short Answer

Step by step solution

Identify the Expressions

Apply the Product of Powers Rule

Add the Exponents

Simplify the Exponent

Key Concepts

Product of Powers Rule

Algebraic Manipulation

Exponents

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