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Chapter 1: Problem 17

Simplify the given expressions. In each of \(5-9\) and \(12-21,\) the result is an integer. $$(\sqrt{5})^{2}$$

Short Answer

Expert verified
The expression \((\sqrt{5})^{2}\) simplifies to 5.

Step by step solution

01

Analyze the Expression

The given expression is \((\sqrt{5})^{2}\). We need to recognize that \(\sqrt{5}\) is the square root of 5.
02

Apply the Square and Square Root Rule

Recall that squaring a square root yields the original number. Therefore, \((\sqrt{5})^{2} = 5\).
03

Final Step: Simplification Result

The expression simplifies directly from \((\sqrt{5})^{2}\) to 5, because the square and square root cancel each other out.

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Key Concepts

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

Square Roots
Square roots are a fundamental part of mathematics, indicating that a number is one of two identical factors that make up a square of the given number. For example, the square root of 5, expressed as \(\sqrt{5}\), is a number which, when multiplied by itself, results in 5. Square roots are crucial for understanding how numbers relate to each other on a deeper level.
  • Positive Roots: In most cases, we consider only the positive root as the principal square root.
  • Visualizing Square Roots: If we think about \(\sqrt{5}\), it represents a number which is between 2 and 3 on the number line.
  • Properties: Understanding the property \((\sqrt{a})^2 = a\) is key in simplifying expressions involving square roots.
Mathematical Rules
Certain mathematical rules play a critical role in simplifying expressions, especially with square roots and powers. It's important to remember these rules as they help streamline complex calculations and inaccurate results.
  • Square and Square Root Relationship: Wise use of the rule \((\sqrt{a})^2 = a\) can simplify expressions quickly by canceling out the operations of taking the square root and squaring.
  • Power of Exponents: The rule \((a^m)^n = a^{m*n}\) assists in managing expressions where numbers are raised to multiple powers.
  • Multiplication Property of Square Roots: Knowing that \(\sqrt{a} \times \sqrt{b} = \sqrt{a\cdot b}\), guides you to handle more complex square roots.
Integer Results
The exercise highlights the concept of integer results when simplifying expressions. In this case, understanding that the expression \((\sqrt{5})^{2}\) simplifies to an integer is key.
  • Defining Integers: Integers are whole numbers and include negative numbers, zero, and positive numbers.
  • Simplification to Integers: Simplifying expressions like \((\sqrt{5})^{2} = 5\) showcases how complex expressions can result in simple, whole numbers.
  • Significance: Understanding when an expression results in an integer can lend clarity and direction, especially in mathematical problem-solving.

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Most popular questions from this chapter

Perform the indicated operations and attach the correct units to your answers. $$\left(238 \frac{\mathrm{kg}}{\mathrm{m}^{3}}\right)\left(\frac{1000 \mathrm{g}}{1 \mathrm{kg}}\right)\left(\frac{1 \mathrm{m}}{100 \mathrm{cm}}\right)^{3}$$

Solve the given problems. The density of water is about \(62.4 \mathrm{lb} / \mathrm{ft}^{3}\). Convert this to kilograms per cubic meter.

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Perform the indicated multiplications. Simplify the expression \(\left(T^{2}-100\right)(T-10)(T+10),\) which arises when analyzing the energy radiation from an object.
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