Problem 77 - Elementary Algebra

Question

\(5^{2}-4 \cdot 1 \cdot 2\)

Step-by-Step Solution

Answer

The answer is 17.

Step 1: Identify the components

Observe the expression given: \(5^{2} - 4 \cdot 1 \cdot 2\). Note that this expression includes exponentiation and multiplication.

Step 2: Calculate the exponentiation

First, calculate the exponentiation: \(5^{2}\). We know that \(5^{2} = 5 \times 5 = 25\). So the expression now becomes: \(25 - 4 \cdot 1 \cdot 2\).

Step 3: Perform the multiplication

Next, perform the multiplication part of the expression: \(4 \cdot 1 \cdot 2\). \(4 \times 1 = 4\), and then \(4 \times 2 = 8\). So the expression now becomes: \(25 - 8\).

Step 4: Perform the subtraction

Finally, perform the subtraction: \(25 - 8 = 17\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number (called the base) is multiplied by itself a certain number of times. The number of times the base is multiplied by itself is determined by the exponent. For example, in the expression \(5^2\), 5 is the base and 2 is the exponent. This means you multiply 5 by itself two times:
\(5^2 = 5 \times 5 = 25\).

Important points about exponentiation:

• The exponent tells us how many times the base is used as a factor.
• A base raised to the power of 2 is also called 'squared', and to the power of 3 is called 'cubed'.
• Any number raised to the power of 1 is the number itself.
• Any number raised to the power of 0 is 1.

Understanding exponentiation is crucial for solving expressions that include exponents, as we always perform exponentiation before other operations.

Multiplication

Multiplication is one of the four basic operations in mathematics. It involves finding the product of two numbers, called factors. In the expression \(4 \times 1 \times 2\), the numbers 4, 1, and 2 are the factors. Multiplication is performed by combining these factors to find their product:
\(4 \times 1 = 4\), and then
\(4 \times 2 = 8\).

Important points about multiplication:

• It is commutative, meaning the order of the factors does not change the product: \(a \times b = b \times a\).
• It is associative, so you can group the factors in any way: \((a \times b) \times c = a \times (b \times c)\).
• Multiplying any number by 1 gives the number itself.
• Multiplying any number by 0 gives 0.

Multiplication is often used in combination with other operations, so it's important to know when to perform it in the sequence of calculations.

Subtraction

Subtraction is the mathematical operation of finding the difference between two numbers. In the expression \(25 - 8\), 25 is called the minuend, and 8 is called the subtrahend. The result of this operation is called the difference:
\(25 - 8 = 17\).

Key points about subtraction:

• Subtraction is not commutative, meaning the order of numbers matters: \(a - b eq b - a\).
• It is not associative: \((a - b) - c eq a - (b - c)\).
• Subtracting 0 from a number leaves it unchanged.
• Subtracting a number from itself gives 0.

Subtraction is typically the final operation when following the order of operations, often coming after exponentiation and multiplication. Understanding its properties ensures accurate results in calculations.