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Chapter 10: Problem 24

Simplify completely. If the radical is already simplified, then say so. $$\sqrt{46}$$

Short Answer

Expert verified
The given expression, \(\sqrt{46}\), is already simplified since there are no perfect squares in its prime factors (2 and 23).

Step by step solution

01

Find the prime factors of 46

First, we need to find the prime factors of the number 46. To do this, we can use the method of dividing the number by its smallest prime factors, starting with 2, and working our way up. 46 is not divisible by 2 (smallest prime number), so we move to the next smallest prime number, which is 3. It's also not divisible by 3. The next smallest prime number is 5, and 46 is not divisible by 5 either. The next smallest prime factor is 7, but 46 isn't divisible by 7. Since we have checked all prime numbers less than the square root of 46, we can conclude that it is a product of two prime numbers; 46=2脳23.
02

Check for perfect squares

Now that we have found the prime factors of 46 (2 and 23), we can check if any perfect squares exist in these factors. Recall that a perfect square is a number that can be expressed as the square of an integer. For example, 4, 9, and 16 are perfect squares since they can be written as 2^2, 3^2, and 4^2, respectively. In this case, none of the prime factors (2 or 23) have an exponent of 2 or higher, which means there are no perfect squares.
03

Conclude the simplification

Since there are no perfect squares in the prime factors of 46, we cannot simplify the radical expression further. Therefore, the given expression, \(\sqrt{46}\), is already simplified.

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Key Concepts

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

Prime Factorization
Prime factorization involves breaking down a composite number into its prime components. This is an essential step in simplifying radicals, as it helps identify any perfect squares within the number. With the number 46, we start by dividing it by the smallest prime numbers. We begin with 2, the smallest prime, and find that 46 is divisible by 2, giving us 23. Thus, 46 can be expressed as the product of these two primes: 46 = 2 脳 23.

  • Start with the smallest prime number.
  • Continue dividing until you reach a prime quotient or can't divide anymore.

Prime factorization is particularly useful for understanding the structure of numbers and is a foundation for further operations like finding the greatest common divisor or least common multiple.
Perfect Squares
Perfect squares are numbers that can be written as the product of an integer with itself. For example, 4 is a perfect square because it equals 2 脳 2. In the context of simplifying radical expressions, identifying perfect squares is crucial because they can be taken out of the square root.

In the prime factorization of 46 (2 脳 23), neither of these factors is a perfect square. Both have an exponent of 1, far from reaching the exponent of 2 needed for a number to be a perfect square. Hence, no further simplification of \(\sqrt{46}\) occurs through the perfect square method.

Understanding perfect squares helps us quickly simplify radicals when possible, making computations easier.
Radical Expressions
Radical expressions are numbers under the sign of a square root, cube root, or other higher roots. Understanding how they work is fundamental to simplifying expressions like \(\sqrt{46}\). It involves ensuring no perfect square factors remain inside the radical symbol. In cases like \(\sqrt{46}\), without perfect squares among its factors, the expression remains as it is.

  • Simplify by removing perfect squares.
  • If none exist, the expression is already simplified.

These principles apply widely, not only in algebra but also in geometry and other areas requiring simplifications of root expressions. This underscores the importance of mastering radical expressions to further delve into more intricate mathematical problems.

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Most popular questions from this chapter

Simplify completely. Assume all variables represent positive real numbers. $$\sqrt[3]{w^{14}}$$

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Fill in the blank. Assume all variables represent positive real numbers. $$\sqrt[5]{16} \cdot \sqrt[5]{7}=\sqrt[5]{2^{5}}=2$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{\sqrt{3}+\sqrt{6}}{\sqrt{2}+\sqrt{5}}$$

Rationalize the denominator and simplify completely. Assume the variables represent positive real numbers. $$\frac{10}{9-\sqrt{2}}$$
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