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Chapter 1: Problem 33

Simplify each exponential expression (leave only positive exponents). $$\frac{1}{n} \sqrt{n^{2}+n^{4}}$$

Short Answer

Expert verified
\((1 + n^2)^{1/2}\)

Step by step solution

01

Write the Expression with Fractional Exponents

Rewrite the square root in terms of exponents to make it easier to simplify. The expression becomes: \[ \frac{1}{n} (n^2 + n^4)^{1/2} \].
02

Factor out the Greatest Common Factor Inside the Square Root

Identify the greatest common factor (GCF) inside the square root. Here, the GCF is \(n^2\). Factor \(n^2\) out of the expression inside the square root: \[ (n^2 + n^4)^{1/2} = (n^2(1 + n^2))^{1/2} \].
03

Simplify the Square Root Expression

Apply the property of square roots \((ab)^{1/2} = a^{1/2} \cdot b^{1/2}\). We have: \[ (n^2(1 + n^2))^{1/2} = (n^2)^{1/2} \cdot (1 + n^2)^{1/2} \].
04

Simplify the Expression Further

Calculate the square root of \(n^2\). This yields \(n\). Thus the expression becomes: \[ n \cdot (1 + n^2)^{1/2} \].
05

Cancel Common Factors

Substitute back into the original expression: \[ \frac{1}{n} \cdot n \cdot (1 + n^2)^{1/2} \]. The \(n\) and \(\frac{1}{n}\) cancel each other, leaving: \[ (1 + n^2)^{1/2} \].
06

Ensure Positive Exponents

The expression \((1 + n^2)^{1/2}\) is already in terms of positive exponents. No further simplification is needed.

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Key Concepts

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

Positive Exponents
Exponents describe how many times a number, known as the base, is multiplied by itself. In expressions involving exponents, you might encounter both positive and negative exponents.
  • **Positive exponents** indicate standard multiplication. For example, \( n^3 \) means \( n \times n \times n \).
  • Negative exponents, such as \( n^{-2} \), mean division or reciprocal: \( \frac{1}{n^2} \).
To work with expressions that contain multiple terms, it's important to ensure that the exponents are positive. This makes the expression easier to interpret and manipulate, as demonstrated in the provided solution. At the end, ensuring all exponents are positive ensures clarity and correctness.
Fractional Exponents
Fractional exponents might initially seem complicated, but they follow straightforward rules related to roots. When an exponent is a fraction, it indicates a root of the number.
  • A fractional exponent like \( n^{1/2} \) corresponds to the square root: \( \sqrt{n} \).
  • The general form is \( n^{a/b} \), which means the \( b \)-th root of \( n^a \).
In our example, the expression \( (n^2 + n^4)^{1/2} \) involves a square root, which is easily rewritten as a fractional exponent. Breaking down expressions using fractional exponents can reveal simpler underlying forms, aiding in their simplification.
Simplifying Expressions
Simplifying expressions is about making an expression as clear and simple as possible. It involves reducing complexity while keeping the expression mathematically correct.
  • Identify and factor out the greatest common factor (GCF) to simplify terms.
  • Apply exponent rules to reduce or combine terms as needed.
  • Cancel out terms where possible to arrive at the simplest form.
In the solution provided, the process of simplifying started with the expression \( \frac{1}{n} (n^2 + n^4)^{1/2} \), factored out \( n^2 \) to simplify inside the square root. The expression was then reduced further by canceling out common factors. Finally, it was ensured that all exponents are positive, resulting in a clean and simple expression: \( (1 + n^2)^{1/2} \). Simplification is a powerful tool for clarifying and solving mathematical problems.

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Most popular questions from this chapter

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