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Chapter 9: Problem 3

Evaluate each radical. $$-\sqrt{\frac{81}{9}}$$

Short Answer

Expert verified
The expression evaluates to -3.

Step by step solution

01

Simplify the Fraction Inside the Radical

First, simplify the fraction \( \frac{81}{9} \). Divide 81 by 9 to get 9. Thus, \( \frac{81}{9} = 9 \).
02

Evaluate the Square Root

Next, take the square root of the simplified fraction. The square root of 9 is 3, therefore \( \sqrt{9} = 3 \).
03

Apply the Negative Sign

Since the original expression is \( -\sqrt{\frac{81}{9}} \), we need to consider the negative sign. Multiply the square root value obtained in Step 2 by -1, giving \( -3 \).

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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 concept in mathematics. They help us understand how to reverse the process of squaring a number. When we say "square root," we're looking for a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by itself equals 9. The symbol for square root is \( \sqrt{} \), so \( \sqrt{9} = 3 \).

It's important to remember:
  • The square root of a perfect square (like 9, 16, or 25) is always an integer.
  • Non-perfect squares result in irrational numbers, which cannot be expressed as simple fractions.
  • When dealing with square roots, understanding the properties of squares and multiplication is crucial.
Simplifying Fractions
Simplifying fractions is the process of making them as simple as possible. This is done by dividing the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD). Let's take the fraction \( \frac{81}{9} \).

To simplify:
  • Identify the greatest number that can divide both 81 and 9 without leaving a remainder. Here, it is 9.
  • Divide the numerator and the denominator by this number: \( 81 \div 9 = 9 \) and \( 9 \div 9 = 1 \).
  • Thus, \( \frac{81}{9} \) simplifies to \( 9 \).

Simplifying helps reduce complexity, especially useful when evaluating square roots or performing further mathematical operations.
Negative Numbers
Negative numbers can seem tricky at first, but they're just numbers less than zero. They are written with a minus sign \(-\) before them. In the expression \(-\sqrt{9}\), the negative sign indicates that we "flip" the positive result of the square root operation.

Key points about negative numbers:
  • When multiplying a positive number by a negative one, the result is negative.
  • When both numbers are negative, multiplying them results in a positive number because a negative times a negative is positive.
  • Negative numbers are used in various real-world contexts, like describing debts or below-zero temperatures.

Understanding how to apply negative signs is essential, particularly when they modify results from operations like square roots.

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Most popular questions from this chapter

For Problems \(55-70\), rationalize the denominators and simplify. All variables represent positive real numbers. $$ \frac{2+\sqrt{3}}{3-\sqrt{2}} $$

Use a calculator to evaluate each radical. (Objective 1) $$\sqrt[3]{9261}$$

Evaluate each radical without using a calculator or a table. (Objective 1) $$-\sqrt{1600}$$

For Problems \(55-70\), rationalize the denominators and simplify. All variables represent positive real numbers. $$ \frac{3}{\sqrt{6}+\sqrt{5}} $$

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt{400}$$
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