Problem 54 Simplify. \(-\sqrt{24}\)... [FREE SOLUTION]

Chapter 7: Problem 54

Simplify. \(-\sqrt{24}\)

Short Answer

Expert verified

-2 \sqrt{6}

Step by step solution

- Factor the number inside the square root

First, factor the number inside the square root: \(24 = 2 \times 2 \times 2 \times 3\). So, we can rewrite the square root as \(\sqrt{24} = \sqrt{2^3 \times 3}\).

- Simplify using properties of square roots

Next, use the property of square roots that states \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). In this case: \(\sqrt{2^3 \times 3} = \sqrt{2^2 \times 2 \times 3} = \sqrt{2^2} \times \sqrt{2 \times 3} = 2 \sqrt{6}\).

- Apply the negative sign

Finally, include the negative sign from the original problem to get \(-\sqrt{24} = -2 \sqrt{6}\).

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Key Concepts

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

Factoring Numbers

Factoring numbers is the process of breaking down a number into its prime components. In other words, you express a number as the product of its prime factors. For example, to factor the number 24, you can write it as:
24 = 2 脳 2 脳 2 脳 3.
Each factor here is a prime number. The prime factors of 24 are therefore 2 (repeated three times) and 3.
Understanding how to factor numbers is essential for simplifying square roots, as it helps you to break down the numbers into forms that are easier to work with.

Properties of Square Roots

The properties of square roots can simplify complex square root expressions. One key property is:
\(\text{For any two positive numbers } a \text{ and } b, \sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).
This property allows us to split square roots into more manageable parts.
For example, \(\(\sqrt{24}\) = \(\sqrt{2^3 \times 3}\) = \(\sqrt{2^2 \times 2 \times 3}\) = \(\sqrt{2^2} \times \sqrt{2 \times 3}\)\).
Simplifying further: \(\sqrt{2^2} = 2\), so \(\sqrt{2^2 \times 2 \times 3}\) = 2\(\sqrt{6}\).
Using these properties makes it easier to work with square roots.

Negative Signs in Algebra

In algebra, negative signs can change the meaning of an expression significantly. A negative sign in front of a square root means that the entire expression is negative.
For example, \(\text{{if the problem is}} \sqrt{24}, \text{{and we simplify it to}} 2\sqrt{6}\), adding a negative sign results in \(-\sqrt{24} = -2\sqrt{6}\).
It's important never to ignore negative signs as they directly impact the value of your expression.
Just like arithmetic, algebra requires careful attention to positive and negative signs to ensure accurate calculations.

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91影视

Simplify. \(-\sqrt{24}\)

Short Answer

Step by step solution

- Factor the number inside the square root

- Simplify using properties of square roots

- Apply the negative sign

Key Concepts

Factoring Numbers

Properties of Square Roots

Negative Signs in Algebra

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