Problem 78 - Elementary Algebra

Question

\(\sqrt{10^{2}-4 \cdot 2 \cdot 8}\)

Step-by-Step Solution

Answer

6

Step 1 - Simplify the expression inside the square root

First, we need to simplify the expression inside the square root. The expression is: \[10^2 - 4 \cdot 2 \cdot 8\].

Step 2 - Calculate the square of 10

Find the value of \(10^2\). \[10^2 = 100\].

Step 3 - Simplify the multiplication

Calculate the multiplication part of the expression: \[4 \cdot 2 \cdot 8 = 8 \cdot 8 = 64\].

Step 4 - Subtract the results

Now subtract the result of the multiplication from the square: \[100 - 64 = 36\].

Step 5 - Find the square root

Calculate the square root of the resulting expression: \[ \sqrt{36} = 6\].

Key Concepts

headline of the respective core concept

The square root is a fundamental concept in mathematics. It is the inverse operation of squaring a number. For any positive number x, the square root of x is the number y such that \(y^2 = x\). This means, if you multiply y by itself, you should get x back. For instance, \(\sqrt{25} = 5\) because \(5^2 = 25\). It is denoted by the radical symbol \(\sqrt{}\). There can be both positive and negative square roots for any given number. However, in most cases, we consider only the principal (positive) square root.

headline of the respective core concept

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. In the given exercise, \(10^2 - 4 \cdot 2 \cdot 8\) is an algebraic expression. To simplify it, you ordinarily need to perform established mathematic operations. \(10^2\), for example, means 10 multiplied by itself, giving 100. The term \( - 4 \cdot 2 \cdot 8\) involves multiplication and subtraction, which aligns with the numerical operations found in algebra.

headline of the respective core concept

Basic arithmetic operations include addition, subtraction, multiplication, and division. They are the building blocks for algebraic expressions and more complicated calculations.

In the exercise, these operations manifest in several steps:

• Calculating 10 squared (\(10^2 = 100\))
• Multiplying 4, 2, and 8 together (\(4 \cdot 2 \cdot 8 = 64\))
• Subtracting the product from the initial squared value (\(100 - 64 = 36\))

These operations collectively reduce the given algebraic expression to a simpler form that can be evaluated further.

headline of the respective core concept

Order of operations is a set rule in mathematics dictating the correct sequence to apply operations when simplifying expressions. It is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures consistency and avoids ambiguity in mathematical expressions.

In the provided problem, the order of operations follows these steps:

• First, calculate any exponents, which is \(10^2\) or 100
• Next, perform multiplication: \(4 \cdot 2 \cdot 8\)
• Finally, address the subtraction: \(100 - 64\)

By following these steps, you ensure that all components are correctly simplified before finding the square root.