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Chapter 10: Problem 5

Simplify each expression as completely as possible. Be sure your answers are in simplest radical form. Assume that all variables appearing under radical signs are nonnegative. $$\sqrt{32}$$

Short Answer

Expert verified
4\sqrt{2}

Step by step solution

01

Identify the Prime Factorization of 32

The first step is to find the prime factorization of the number inside the square root. The number 32 can be expressed as a product of prime factors: 32 = 2 脳 2 脳 2 脳 2 脳 2 or 2^5.
02

Rewrite 32 Under the Square Root Using Its Prime Factorization

\[\sqrt{32} = \sqrt{2^5}\]
03

Simplify the Square Root Expression

Group the factors in pairs and simplify: \[\sqrt{2^5} = \sqrt{(2^2) \cdot (2^2) \cdot 2} = \sqrt{(2^2) \cdot (2^2)} \cdot \sqrt{2} = 2 \cdot 2 \cdot \sqrt{2} = 4\sqrt{2}\]

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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 breaking down a number into its prime factors. A prime factor is a number that is only divisible by 1 and itself. For example, the prime numbers are 2, 3, 5, 7, etc. To find the prime factorization of 32, you divide 32 by the smallest prime number, which is 2, and continue dividing the quotient by 2 until you reach 1. This gives us:
  • 32 梅 2 = 16
  • 16 梅 2 = 8
  • 8 梅 2 = 4
  • 4 梅 2 = 2
  • 2 梅 2 = 1
So, the prime factorization of 32 is 2 脳 2 脳 2 脳 2 脳 2 or 2^5. Understanding prime factorization is essential for simplifying square roots.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because 4 脳 4 = 16. Mathematically, we represent the square root using the symbol 鈭�. When you see 鈭�32, it means you are looking for a number which, when squared, equals 32. Simplifying square roots involves prime factorization and breaking down the expression into simpler forms. Remember that 鈭�(a 脳 b) = 鈭歛 脳 鈭歜. This property is key when working with square roots.
Simplifying Expressions
Simplifying expressions means to rewrite them in their simplest form. For the given problem 鈭�32, we use prime factorization to express 32 as 2^5. Then we can group the prime factors in pairs:
  • 32 = 2 脳 2 脳 2 脳 2 脳 2
We can rewrite this under the square root as 鈭�(2^5). The next step is to identify pairs of numbers under the square root:
  • 鈭�(2^5) = 鈭�((2^2) 脳 (2^2) 脳 2)
Since 鈭�(a 脳 a) = a, we can further simplify by taking the square root of the pairs:
  • 鈭�((2^2) 脳 (2^2) 脳 2) = 2 脳 2 脳 鈭�2 = 4鈭�2
Therefore, the simplified form of 鈭�32 is 4d鈭�2.
Radicals
Radicals are expressions that contain a root, such as a square root, cube root, or n-th root. The most common radical is the square root, denoted by 鈭�. Radicals can often be simplified by removing squares from underneath the root. For example, to simplify,鈭�32, we use the fact that 鈭歛 is simplified by finding any square factors of a. By using prime factorization, we determined that 32 = 2^5, and identified pairable squares (2^2). Thus, 鈭�32 simplifies to 4鈭�2. Working with radicals requires practice and understanding their properties, such as:
  • 鈭�(a 脳 b) = 鈭歛 脳 鈭歜
  • 鈭�(a^2) = a
Recognizing these properties helps in breaking down and simplifying complex radical expressions.

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Most popular questions from this chapter

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{12}+\sqrt{27}$$

Use a calculator to find the following square roots. Round off your answers to the nearest hundredth. $$\sqrt{730}$$

Estimate the given square root between two consecutive integers without using a calculator, then use a calculator to find the square root rounded to two decimal places to confirm your estimate. $$\sqrt{115}$$

Rationalize the denominators and simplify. $$(3+\sqrt{5})^{2}+\frac{8}{3-\sqrt{5}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{28}-\sqrt{7}$$
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