Problem 38 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \((4 \cdot 9)^{3}\)

Step-by-Step Solution

Answer

The simplified form of the expression is 46656.

Step 1: Apply brackets

The expression inside the brackets is \(4 \cdot 9\), which simply means multiplying 4 by 9. The result is 36.

Step 2: Apply Exponent

After the multiplication, the expression now becomes \(36^{3}\). This simply means multiplying 36 by itself three times.

Step 3: Simplification

Multiply 36 by itself three times. \(36 \cdot 36 \cdot 36 = 46656\).

Key Concepts

Order of Operations

When faced with a mathematical expression like ewline ewline ewline ewline that combines different operations, such as multiplication and exponentiation, knowing the correct sequence to tackle these operations is crucial for arriving at the right answer. This sequence is known as the order of operations. In mathematics, the order in which we perform operations is determined by the rules memorized using the acronym PEMDAS or BODMAS. These stand for Parentheses or Brackets, Exponents or Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the exercise ewline ewline ewline ewline , you first deal with the operation within the parentheses (multiplication), before proceeding to the exponentiation. This adherence to the order of operations ensures the expression is simplified correctly to arrive at the final solution.

Exponentiation

Exponentiation is an operation that raises a number, known as the 'base', to the power of an 'exponent'. In layperson's terms, it tells you how many times to multiply the base by itself.

For example, in the expression ewline ewline ewline ewline , the base is 36 and the exponent is 3, signifying that you should multiply 36 by itself two additional times (three times in total). It is important to note that exponentiation is not the same as multiplication; it's a form of repeated multiplication that applies to a base number. Taking into account this practice can make long calculations like those involving exponential growth or compounding interest much simpler.

Multiplication

Multiplication, one of the four basic operations in arithmetic, involves combining equivalent groups. When the expression ewline ewline ewline ewline is given, the first operation we perform is multiplying 4 by 9 to consolidate these groups into a single number. This results in 36, a process akin to saying that there are four groups of nine items each, ending up with a total of 36 items.

The concept of multiplication is not limited to small numbers. In the exercise step where ewline ewline ewline ewline you're essentially calculating the volume of a cube with a side length of 36, which involves three dimensional multiplications. Despite being a more complex form of multiplication, the fundamental idea remains the same - you are combining multiple groups of the same size.