Problem 63 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Give the exact value if possible. Otherwise, approximate to the nearest hundredth. $$ -\sqrt{20} $$

Step-by-Step Solution

Answer

The result of the expression is approximately \(-4.47\).

Step 1: Identify the Negative Square Root

First, note the negative sign in front of the square root, \( -\sqrt{20} \). This means the final answer will be negative.

Step 2: Calculate the Square Root

Next, compute the square root of the number 20 inside the radical sign. Since 20 is not a perfect square, its square root will be an irrational number. Using a calculator, \(\sqrt{20}\) approximates to 4.47.

Step 3: Apply the Negative

Finally, apply the negative sign to the square root of 20. The result is \(-4.47\).

Key Concepts

Understanding Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction or a ratio of two integers. They have non-repeating, non-terminating decimal expansions. When dealing with square roots, irrational numbers typically occur when we take the square root of a number that is not a perfect square, like 20. This is because the square root cannot be simplified into an exact fraction.

Examples of irrational numbers include:

• π (Pi), which represents the ratio of the circumference to the diameter of a circle
• e, the base of the natural logarithm
• √2, the square root of two

While some numbers, like 25, have perfect square roots (like 5), others, such as 20, do not easily resolve into a neatly rounded number, leading to an irrational result.

Evaluating Expressions Involving Square Roots

Evaluating expressions with square roots involves simplifying the root when possible and applying any coefficients or constants involved. In the exercise, when evaluating an expression such as the key steps are:

• Identify any signs (positive or negative) before the square root.
• Simplify the square root as much as possible. If the number inside the square root is a perfect square, it should be simplified.
• If it is not a perfect square, the result will be irrational, which means approximations might be needed.
• Apply any constants or coefficients outside the radical sign to the evaluated root.

Understanding these steps will help in accurately solving expressions that involve square roots, ensuring that both exact values are found when possible, or approximate values when necessary.

Approximating Square Roots

Since not all numbers are perfect squares, approximating square roots becomes necessary. A perfect square results in a rational number. For instance, the square root of 9 is 3. However, non-perfect squares like 20 yield irrational numbers, requiring approximations. Here’s how to approximate square roots:

• Identify two perfect square numbers between which the target number lies. For 20, these are 16 (because the square root of 16 is 4) and 25 (since the square root of 25 is 5).
• Realize that since 20 is closer to 16 than 25, its square root will be closer to 4.
• Use a calculator for a more precise approximation: he value of to 4.47.

By understanding these methods, you can approximate square roots manually or with the help of technology, giving you greater flexibility in solving math problems involving irrational numbers.