Problem 7 - Algebra Form and Function

Question

Find the sequence of operations to undo the sequence given. Raise to the \(5^{\text {th }}\) power and then multiply by 2 .

Step-by-Step Solution

Answer

Answer: To undo the given sequence of operations, first divide the number by 2, and then raise the result to the 1/5 power. In mathematical notation, for an input number x, perform the operation (x/2)^(1/5).

Step 1: Analyze the given sequence and its operations

The given sequence consists of two operations: (1) raise a number to the 5th power, and (2) multiply the result by 2. To undo these operations, we need to reverse the order of the operations and then perform the inverse operations.

Step 2: Reverse the order of the operations in the given sequence

Reversing the order of operations in the given sequence gives us the new order of operations: (1) divide by 2, and (2) raise to the \(1/5\) power.

Step 3: Apply the inverse operations to an unknown number

Let's denote the unknown number as x. We will apply the inverse operations, which are dividing by 2 and raising to the \(1/5\) power, in the correct order. First, we divide x by 2: \(x/2\). Next, we raise the result to the \(1/5\) power: \((x/2)^{1/5}\).

Step 4: Write the final sequence of operations to undo the given sequence

The sequence of operations to undo the given sequence is as follows: First, divide the number by 2, and then raise the result to the \(1/5\) power. In mathematical notation, for an input number x, perform the operation \((x/2)^{1/5}\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They take the form: \( a^x \), where \( a \) is the base and \( x \) is the exponent. These functions are crucial in modeling growth and decay processes, such as population growth or radioactive decay. The base of the exponent determines the function's growth rate. For instance, bases greater than one result in exponential growth, while those between zero and one cause exponential decay.Raising a number to a power larger than one, such as 5, which was in the original exercise, indicates that the number will be multiplied by itself repeatedly—five times in this case. Understanding how this operation works is vital for correctly applying or undoing such operations, as seen in various scientific and financial calculations.Moreover, exponential operations have specific inverse operations known as roots. For instance, raising a number to the \(1/5\) power is the inverse of raising it to the 5th power. These inverse operations can help solve equations by reversing previous exponential actions.

Order of Operations

The order of operations is a set of rules that determine the sequence in which calculations are performed. This system is crucial to solving math problems accurately and consistently. It is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents (including roots, like the \(1/5\) power in our exercise)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the given exercise, understanding the correct order helps in determining how to properly apply inverse operations. In the sequence of operations given, we first raise a number to a power and then multiply. To undo this process, we need to reverse the order, first dividing, then raising to a power.Reversing requires careful attention to the order, ensuring everything is correctly undone. Misapplying these rules can lead to errors, especially in solving equations or simplifying expressions.

Mathematical Notation

Mathematical notation provides a universal language that conveys complex ideas succinctly. It includes symbols and equations that describe mathematical operations, relationships, and functions.In the step-by-step solution, the notation \((x/2)^{1/5}\) illustrates how we undo the original sequence of operations. This notation shows two critical concepts:

• The division of \(x\) by 2, simplifying or reversing multiplication by 2.
• Raising the result to the \(1/5\) power, which is the inverse of the \(5^{th}\) power.

Using this notation makes instructions clear and systematic, which is especially important for complex calculations. The ability to represent these calculations using concise symbols prevents misunderstanding and provides an efficient way to communicate mathematical ideas across different contexts and languages.Understanding how to interpret and write mathematical notation is a necessary skill when learning to work with complex sequences of operations or functions, as demonstrated in our exercise.