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Chapter 10: Problem 36

simplify each expression. $$\sqrt{(-10)^{2}}$$

Short Answer

Expert verified
The simplified expression is 10

Step by step solution

01

Understanding the Order of Operations

Recall the order of operations, popularly known by the acronym PEMDAS, standing for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). According to this rule, calculations inside parentheses are done first, followed by exponents.
02

Execute the Expression inside the Parentheses

(-10)^{2} is the square of -10. The square of any real number, whether it's positive or negative, is always positive because the two like signs become positive when multiplied. Therefore, the square of -10 is 100.
03

Calculate the square root

The next step is to take the square root of the result from Step 2. The square root of 100 is 10, because 10 * 10 is 100.

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Key Concepts

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

Order of Operations
When solving algebraic expressions, it's crucial to follow the correct order of operations. This order ensures that calculations are consistent and correct, regardless of who is solving the problem. The widely used acronym PEMDAS can help you remember the sequence:
  • Parentheses (solve expressions within parentheses first)
  • Exponents (next, handle any exponents or powers)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (also from left to right)
Understanding PEMDAS helps you confidently tackle expressions like \((-10)^2\), ensuring you square the negative number before moving on to further steps. It's like having a reliable roadmap guiding each calculation to its correct destination.
Exponents
Exponents are a fundamental concept in algebra used to express repeated multiplication of a number by itself. When we encounter an expression like \((-10)^2\), it means \(-10 \times -10\). Let's break it down:
  • The negative sign before 10 stays with the number being squared, so it鈥檚 crucial in determining the answer.
  • Squaring \(-10\) results in positive 100. This illustrates the rule that squaring a negative number yields a positive result, as multiplying two negative numbers together results in a positive product.
By mastering the concept of exponents, especially squares, you'll find that simplifying even complex expressions becomes more manageable.
Square Roots
The square root is the operation to find a number that, when multiplied by itself, gives a certain product. For example, the square root of 100 is 10, because \(10 \times 10 = 100\). To determine the square root of a number efficiently:
  • Start with knowing that it's looking for a number which, when squared, results in the original number.
  • The symbol for square root is \(\sqrt{}\).
    • For \([\sqrt{100}]\), it simplifies directly to 10.
  • Note that the square root of a number asks for the principal root, which is the non-negative root.
Appreciating the properties of square roots can simplify solving expressions significantly, especially when dealing with perfect squares like 100, which yield an integer result.

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Most popular questions from this chapter

Solve: \(3 x-4 \leq 2\) and \(4 x+5 \geq 5\) (Section 9.2, Example 2)

Explain how to perform this multiplication: \(\sqrt{2}(\sqrt{7}+\sqrt{10})\)

In Exercises \(93-104\), rationalize each numerator. Simplify, if possible. $$\frac{\sqrt{x+5}-\sqrt{x}}{5}$$

In Exercises \(39-64,\) rationalize each denominator. $$\sqrt[3]{\frac{3}{x y^{2}}}$$

In Exercises \(105-112,\) add or subtract as indicated. Begin by rationalizing denominators for all terms in which denominators contain radicals. $$\sqrt{6}-\sqrt{\frac{1}{6}}+\sqrt{\frac{2}{3}}$$
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