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Chapter 9: Problem 4

Evaluate each radical without using a calculator or a table. (Objective 1) $$-\sqrt{36}$$

Short Answer

Expert verified
The value of \(-\sqrt{36}\) is \(-6\).

Step by step solution

01

Identify the Radical

The given expression is \(-\sqrt{36}\). Our task is to evaluate the square root of 36 and then apply the negative sign.
02

Evaluate the Square Root

To evaluate \(\sqrt{36}\), we need to find a number which when multiplied by itself results in 36. The number that satisfies this condition is 6, because \(6 \times 6 = 36\). So, \(\sqrt{36} = 6\).
03

Apply the Negative Sign

Since the expression has a negative sign in front of the square root, we need to apply it to the result from Step 2. This results in \(-6\).
04

Final Evaluation

After applying the negative sign, we conclude that the evaluation of \(-\sqrt{36}\) is \(-6\).

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Key Concepts

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

Understanding Square Roots
The square root of a number is essentially the mathematical operation that reverses the process of squaring. Squaring a number means multiplying it by itself.
For example, in the case of 36, we think of the number that, when squared (multiplied by itself), gives us 36. This number is 6 because \(6 \times 6 = 36\).
  • Square roots and squares are opposite operations.
  • If \( a^2 = b \), then \( \sqrt{b} = a \).
It's important to note that every positive number has two square roots: one positive and one negative. In mathematical notation, the square root symbol \(\sqrt{\cdot}\) by default refers to the positive (or principal) square root.
Exploring Mathematical Evaluation
Mathematical evaluation involves simplifying an expression to reach a single, straightforward answer. It consists of processes or steps that follow mathematical rules to solve a problem.
The expression \(-\sqrt{36}\) asks us to evaluate or calculate the value. Here is how we evaluate it step-by-step:
  • First, we address the operation inside the radical. In other words, we determine \(\sqrt{36}\), which is 6.
  • The next step is to then apply any additional operations, which is the negative sign in this case, resulting in \(-6\).
Evaluation requires attention to both the numbers and the operations used, ensuring each step aligns with mathematical standards.
Interpreting Negative Numbers in Radicals
Negative numbers can often be confusing when they appear in front of radicals, but they follow clear mathematical principles. In the expression \(-\sqrt{36}\), the negative sign is not under the square root; it applies outside the square root operation.
  • The negative sign modifies the outcome after the radical is solved, not the number under the radical.
  • Negative products: Whenever you see a negative sign in front of a number, it just flips the number's sign (e.g., the result of 6 is flipped to \(-6\)).
This means we first find \(\sqrt{36} = 6\), and subsequently apply the negative, altering the result to \(-6\). Negative numbers in this context signify direction or position on the number line, rather than implying a change in value to the contents of the square root.

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Most popular questions from this chapter

Simplify each expression by using the distributive property. (Objective 2) $$7 \sqrt{5}-9 \sqrt{6}+14 \sqrt{5}-2 \sqrt{6}$$

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt{9801}$$

Use a calculator to find a rational approximation of each square root. Express your answers to the nearest hundredth. $$\sqrt{98}$$

Simplify each expression by using the distributive property. (Objective 2) $$17 \sqrt{7}-9 \sqrt{7}$$

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt{49}$$
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