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Chapter 1: Problem 9

Simplify the given expressions. In each of \(5-9\) and 12-21, the result is an integer. $$-\sqrt{49}$$

Short Answer

Expert verified
The simplification of \(-\sqrt{49}\) is -7.

Step by step solution

01

Identify the Square Root

The expression given is \(-\sqrt{49}\). First, we need to identify the square root of the number under the radical, which is 49.
02

Calculate the Square Root

Calculate the square root of 49. Since 49 is a perfect square, its square root is 7. Thus, \(\sqrt{49} = 7\).
03

Apply the Negative Sign

The expression has a negative sign in front of the square root. So, we need to multiply the square root result by -1. Therefore, \(-\sqrt{49} = -7\).

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Key Concepts

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

Square Roots
The concept of square roots revolves around finding a number which, when multiplied by itself, gives the original number under the root. For instance, the square root of 49 is 7, because multiplying 7 by itself results in 49. Square roots are often represented with the radical symbol \(\sqrt{}\). Let's see how to approach square roots in various scenarios:
  • **Find the Base Number:** Begin by recognizing that you are trying to determine which number, when squared (multiplied by itself), will give the original number under the symbol.
  • **Perfect Squares Simplify to Integers:** If the number under the radical is a perfect square (like 49), the square root will result in an integer. This is the most straightforward scenario.
  • **Tip in Calculation:** If a calculator isn't handy, remember that numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are perfect squares, and you can memorize a few of these to speed up calculations.
Approaching square roots with these steps can make problems more straightforward and less intimidating, even when the expressions become more complex.
Negative Numbers
Negative numbers are numbers that are less than zero, typically indicated by a minus sign (-) in front of them. In the context of square roots and expressions, it's crucial to follow these principles to correctly handle negative numbers:
  • **Negative Before or Inside the Root?**: Numbers can become negative before or after calculating a square root. If a negative sign is outside the \(\sqrt{}\), it implies the result should be negative like in \(-\sqrt{49} = -7\).
  • **Oddities with Negative Inside the Root**: Generally, square roots of negative numbers are not defined in terms of real numbers. For real numbers, the square root value itself is always positive or zero.
  • **Conceptual Understanding:** Grasping how negatives impact an expression can help simplify and correctly align terms in broader expressions.
Dealing with negative numbers requires careful attention to where the negative sign is positioned relative to mathematical operations.
Perfect Squares
Perfect squares are numbers obtained by squaring a whole number. For example, 49 is a perfect square because it is the result of multiplying 7 by itself. Recognizing perfect squares can simplify problems because they directly yield whole number roots.
  • **Faster Calculations**: Knowing perfect squares allows for quicker recognition of square roots without relying on calculators.
  • **Pattern Recognition**: Familiarity with perfect squares such as 1, 4, 9, 16, etc., aids in quickly resolving values when they appear under radical signs.
  • **Simplification**: When simplifying expressions, confirming that numbers are perfect squares can lead to immediate answers. In our example, identifying 49 as a perfect square simplifies \(-\sqrt{49}\) to \(-7\) quickly.
Utilizing perfect squares can not only streamline solving equations but also provide a solid foundation in understanding number relationships.

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