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Chapter 10: Problem 23

Simplify completely. If the radical is already simplified, then say so. $$\sqrt{38}$$

Short Answer

Expert verified
The given square root is already in its simplest form: \(\sqrt{38}\)

Step by step solution

01

Identify the given square root

Given square root is: \[\sqrt{38}\]
02

Prime factorization of the number under the square root

We will find the prime factors of 38: 38 can be divided into its prime factors as 2 * 19.
03

Check for any possible simplification

Since 38 has two distinct prime factors (2 and 19), no further simplification can be done for this square root.
04

Write the final answer

The given square root is already in its simplest form: \[\sqrt{38}\]

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Key Concepts

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

Prime Factorization
Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves. For example, 2, 3, 5, 7, 11 are all prime numbers.

When you factorize a number, you break it down into its prime components. This method is especially useful when dealing with square roots. In the case of the number 38, we find its prime factors to help simplify its square root if possible.
  • To factor 38, we first look for the smallest prime number that divides 38, which is 2.
  • Dividing 38 by 2 gives us 19. Since 19 is also a prime number, the factorization of 38 is complete, giving us 2 and 19 as its prime factors.
Understanding prime factorization helps when simplifying square roots and performing other arithmetic operations on numbers.
Radicals
A radical is a symbol that represents the root of a number. The most common radical is the square root, denoted by the symbol \(\sqrt{}\). The expression inside the radical is called the radicand. In our example, \(\sqrt{38}\), 38 is the radicand.

Radicals can be simplified using the prime factorization method. If a number inside the radical is a perfect square, or a multiple of a perfect square, it can be simplified. However, not all radicals can be simplified neatly into whole numbers.
  • If the radicand has perfect square factors, you can extract them from under the radical.
  • If no such factors exist, like in \(\sqrt{38}\), the square root remains in its original form.
When working with radicals, it's important to determine whether the radical can be simplified for easier calculations or if it is already in its most reduced form.
Simplest Form
The simplest form of a radical expression is when the radical cannot be reduced further. This means that there are no more perfect square factors in the radicand that can be extracted.

In the case of \(\sqrt{38}\), following the steps of prime factorization, we found that 38 is made up of the prime numbers 2 and 19. Since neither of these are perfect squares, and they do not combine to form a perfect square, the radical is considered to be in its simplest form.
  • To simplify, always check if the radicand can be broken down into factors that are perfect squares.
  • In instances where no simplification is possible, acknowledge the radical as already being at its simplest form.
Understanding when a radical is in its simplest form helps in managing calculations accurately and efficiently.

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Most popular questions from this chapter

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{\sqrt{8}}{\sqrt{3}+\sqrt{2}}$$

Rationalize the denominator of each expression. Assume all variables represent positive real numbers. $$\sqrt[4]{\frac{10}{27}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{\sqrt{32}}{\sqrt{5}-\sqrt{7}}$$

Simplify completely. $$\frac{18-6 \sqrt{7}}{6}$$

What does it mean to rationalize the denominator of a radical expression?
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