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Chapter 8: Problem 11

Fill in the blanks. Integers such as \(4,9,\) and 25 are called ____ Squares.

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Step by step solution

01

Identify the type of numbers

First, analyze the list of integer values given: \(4, 9, and 25\). The square root of each of these numbers results in a non-decimal and non-fractional value. \(2^2 = 4, 3^2 = 9, 5^2 = 25\). Therefore, these specific integers are called perfect squares.

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Key Concepts

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

Integers
Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. They do not have any fractional or decimal components. Integers are fundamental in mathematics because they are used to count, order, and label. Here are some key points to understand about integers:
  • The set of integers is denoted by the symbol \( \mathbb{Z} \).
  • Examples of integers include \(-5, 0, 3,\) and \(42\).
  • Integers can be used in various operations such as addition, subtraction, multiplication, and division.
Understanding integers is essential for solving problems in mathematics, as they are often the basis for more complex numbers and mathematical concepts. Whether you're balancing a budget or measuring a distance, integers are often involved.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. The square root symbol is \( \sqrt{} \). It's important to know that not all integers have whole number square roots.

Here鈥檚 what you need to know about square roots:
  • The square root of a perfect square is always an integer. For example, \( \sqrt{25} = 5 \) because \( 5 \times 5 = 25 \).
  • Square roots of non-perfect squares are often irrational numbers, meaning they cannot be expressed as a simple fraction.
  • Solving the square root of negative numbers introduces the concept of imaginary numbers.
When learning about square roots, it's beneficial to start with perfect squares, as they introduce the concept with numbers that have neat and tidy solutions. Perfect squares like \(4, 9, 16,\) and \(25\) are great examples to practice finding square roots.
Mathematics Vocabulary
Mathematics has its own language, consisting of terms and expressions that help convey complex ideas. Understanding math vocabulary is crucial for mastering concepts. In the context of perfect squares:
  • Perfect Squares: These are integers that can be expressed as the square of another integer. For instance, \(9\) is a perfect square because \(3^2 = 9\).
  • Non-negative: Perfect squares are always non-negative because squaring a positive or negative integer yields a positive result.
  • Root: This term refers to the number that, when multiplied by itself, reaches the original value. For example, the square root of \(16\) is \(4\).
A solid grasp of math vocabulary allows students to easily navigate problems and communicate mathematical reasoning effectively. It helps in accurately interpreting questions and providing clear solutions.

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Most popular questions from this chapter

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{245}{36}}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\frac{q^{3 / 4} q^{4 / 5}}{q^{-2 / 3}}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\frac{3}{4} \sqrt{192 a^{3} b^{5}}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\frac{1}{5} x^{5} y \sqrt{75 x^{3} y^{2}}$$

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$\left(\frac{9}{64}\right)^{1 / 2}$$
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