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Chapter 0: Problem 42

Find each square root. If it is not a real number, say so. $$ \sqrt{64} $$

Short Answer

Expert verified
The square root of 64 is 8.

Step by step solution

01

- Identify the given number

The number given in the exercise is 64, and we need to find its square root.
02

- Determine if the number is a perfect square

Check if 64 is a perfect square by finding a number that, when multiplied by itself, equals 64.
03

- Perform the square root operation

Calculate the square root of 64. The square root of 64 is 8 because \(8 \times 8 = 64\).
04

- Verify if the square root is a real number

Since 8 is a real number, \(\sqrt{64} = 8\).

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Key Concepts

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

perfect squares
When we talk about perfect squares, we mean numbers that can be expressed as the product of an integer with itself. For example, 64 is a perfect square because it equals 8 multiplied by 8 \( \sqrt{64} = 8\).
Other common examples of perfect squares include:
  • 1 (since 1 \( \times \ 1 = 1\))
  • 4 (since 2 \( \times \ 2 = 4\))
  • 9 (since 3 \( \times \ 3 = 9\))
  • 16 (since 4 \( \times \ 4 = 16\))
Recognizing perfect squares can make math easier because you know their square roots immediately. It's important to spot these squares so you can quickly compute or validate the square root operation.
real numbers
In mathematics, real numbers include both rational and irrational numbers. They cover every number that you can find on the number line.
Examples of real numbers include:
  • Integers like -2, 0, 3
  • Fractions like 1/2, 3/7
  • Irrational numbers like \( \pi \) and \( \sqrt{2} \)
Real numbers are contrasted with imaginary numbers, which involve the square roots of negative numbers.
For example, the square root of -1 is represented by i, where i is an imaginary unit. In the context of the provided exercise, the square root of 64 is 8, which is a real number.
If you encounter an exercise where the square root of a negative number is needed, the result is not a real number but an imaginary number.
square root operation
The square root operation involves finding a number that, when multiplied by itself, gives the original number. This is often represented using the radical symbol ( \sqrt{} \ ).
Steps to find a square root:
  • Identify the number you need to find the square root of.
  • Determine if it's a perfect square, this simplifies the process.
  • If it is a perfect square, find the integer that when multiplied by itself results in the original number.
  • If it's not a perfect square, you may need to use estimation or a calculator.
For example, in the exercise, you needed to find \sqrt{64}. You identified 64 is a perfect square because \(8 \times 8 = 64\), thus \( \sqrt{64} = 8 \).
It's useful to be familiar with at least the first 15 perfect squares (like 1, 4, 9, up to 225) to quickly perform square root operations.

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Most popular questions from this chapter

Simplify each expression. $$ 9 \cdot 4-8 \div 2 $$

Evaluate each expression for \(w=4, x=-\frac{3}{4}, y=\frac{1}{2},\) and \(z=1.25 .\) $$ x y+y^{4} $$

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Write each percent as a fraction. Give answers in lowest terms. See Example 13. $$ 140 \% $$

Simplify each inequality if needed. Then determine whether the statement is true or false. $$ -7<4+1 $$
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