Problem 79 - Elementary Algebra

Question

\(\sqrt{2^{2}-4(-2)(4)}\)

Step-by-Step Solution

Answer

6

Step 1: Identify the equation

The given equation is \ \ \ \( \sqrt{2^{2}-4(-2)(4)} \ \ \).

Step 2: Simplify inside the square root

First, solve the expression inside the square root: \ \ \(2^{2}-4(-2)(4)\ \ \).

Step 3: Calculate exponents and products

Calculate the square of 2 and the product of \-4, \-2, and 4: \ \ \(2^{2} = 4 \ \ \) and \ \ \((-4) \cdot (-2) \cdot 4 = 32\ \ \ \).

Step 4: Combine the results

Now add the results: \ \ \(4 + 32 = 36\ \ \ \).

Step 5: Take the square root

Finally, take the square root of 36: \ \ \(\sqrt{36} = 6 \ \ \).

Key Concepts

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 36 is 6, because \(6 \times 6 = 36\). To simplify a square root, you usually need to identify any perfect squares inside the radical symbol. Some important things to remember about square roots:

• The square root of a positive number has two solutions: a positive and a negative, because both will square to the original number. For instance, \( \text{if} \sqrt{36} = 6, \text{then} \sqrt{36} = -6\) also holds true since \(-6 \times -6 = 36\).
• Square roots only apply to non-negative numbers in basic real number arithmetic.

Understanding how square roots work is essential for solving problems in algebra, especially those involving radicals and exponents.

Exponents

Exponents are used to express repeated multiplication of the same number. For instance, \(2^2\) means \(2 \times 2 = 4\). Here are some important points about exponents:

• An exponent tells you how many times to multiply the base number by itself.
• Common terms include 'squared' (raised to the power of 2) and 'cubed' (raised to the power of 3).
• Exponents can also be negative or fractions, but those are more advanced topics.

In our example, we used \(2^2 = 4\) to simplify part of the expression inside the square root.

Basic Algebra

Basic algebra involves using symbols to represent numbers and finding the value of these symbols. Here's a quick look at some fundamental aspects of algebra:

• Variables: Symbols (often letters) that stand in for unknown values.
• Expressions: Combinations of variables, numbers, and operations (like addition or multiplication).
• Equations: Statements that two expressions are equal, often used to find the value of variables.

In our example, we had an expression inside the square root. We simplified this expression step by step to find the final answer.

Radicals

Radicals involve taking roots of numbers. The most common radical is the square root, but there are also cube roots, fourth roots, etc. Key aspects to remember about radicals:

• The radical symbol (\(\sqrt{}\)) denotes the root.
• Radical expressions can often be simplified by factoring out perfect squares (or other roots).
• Operations with radicals follow specific rules, like combining or breaking down radicals using multiplication or division.

In our exercise, simplifying the radical was the final step where we calculated \(\sqrt{36} = 6\). Understanding radicals helps in breaking down complex expressions into simpler parts.