Problem 64 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. $$2 \cdot\left\\{6+\left[10^{2}-6 \sqrt{25}\right]\right\\}$$

Step-by-Step Solution

Answer

The result is 152.

Step 1: Simplify Inside the Square Root

First, simplify the expression inside the square root: \( \sqrt{25} = 5 \). The square root of 25 is 5.

Step 2: Subtract the Square Root Result

Substitute the result from Step 1 into the expression:\[10^2 - 6\sqrt{25} = 100 - 6 imes 5\]Now simplify the multiplication first: \( 6 \times 5 = 30 \). Next, subtract this from 100: \( 100 - 30 = 70 \).

Step 3: Simplify Inside the Curly Braces

Substitute the result of the subtraction into the expression inside the curly braces:\[6 + [10^2 - 6\sqrt{25}] = 6 + 70\] Add 6 and 70 together: \( 6 + 70 = 76 \).

Step 4: Multiply by 2

Finally, multiply the result from Step 3 by 2:\[2 \cdot \{6 + [10^2 - 6\sqrt{25}]\} = 2 \times 76 = 152\]This is the result of the given expression.

Step 5: Verify with a Calculator

Enter the expression exactly as given into a calculator: \(2 \cdot (6 + ( 10^2 - 6 \cdot \sqrt{25} )) \) and verify that it equals 152.Both manual calculation and the calculator confirm the same result.

Key Concepts

Square Root

The square root is a mathematical operation that finds the number that, when multiplied by itself, gives the original number. In simpler words, if you know that \( x^2 = y \), the square root of \( y \) is \( x \).
To find the square root of a number, such as 25, ask yourself: "What number times itself equals 25?" The answer is 5 because \( 5 \times 5 = 25 \). Breaking it down further, if you were to look at a perfect square like 25, you'll understand that it fits into this explanation perfectly. So, when you encounter \( \sqrt{25} \), the result is 5.

• Square Root of 25 is 5
• The square identifies numbers with whole values

This operation simplifies expressions and often appears in brackets or parentheses in complex problems. Always resolve the square root first when dealing with order of operations.

Exponentiation

Exponentiation refers to the operation of raising a number to a power, which means multiplying the number by itself a specified number of times. The notation \(10^2\) reads as "10 raised to the power of 2" or "10 squared."
Let's take a closer look at this step: \(10^2 = 10 \times 10 \). Therefore, it equals 100. Exponentiation significantly influences calculations by escalating the value of numbers involved.
It’s a powerful mathematical operation that is easily mistaken by its complexity but is actually straightforward if you tackle each part step-by-step.

• Determine the base number (in this case, 10)
• Look at the exponent number (2 here means multiply the base by itself)
• Calculate the resulting value, \(10 \times 10 = 100\)

Multiplication and Division

In mathematics, multiplication and division are operations that simplify numbers further according to the given operations. When doing these calculations, always work from left to right.
In the provided exercise, after calculating the components inside the brackets, you face the multiplication: \(6 \times 5\).

• Identify the numbers to multiply: Here, it was 6 and 5.
• Multiply them to get the answer: \(6 \times 5 = 30\).

This step is crucial as it sets the stage for subtraction or any further operation required in the problems you are solving. Multiplication and division operators share the same precedence, meaning they are resolved as they appear from left to right.

Parentheses and Brackets

Parentheses and brackets help in organizing operations and indicating which calculations to perform first in mathematical expressions. They take precedence in the order of operations. This precedence ruleshow the hierarchy should be followed, often remembered as PEMDAS/BODMAS.

• P: Parentheses first
• E (followed by brackets in BIDMAS): Exponents (ie Powers and Square Roots, etc.)
• MD: Multiplication and Division (left to right)
• AS: Addition and Subtraction (left to right)

Use this rule to solve complex math expressions accurately. In our context, the brackets guide you to resolve each nested calculation step by step, beginning with any operations inside the square brackets, then the curly braces, before handling the outer expression.