Question

State in words the inverse operation. Take the ninth root.

Short answer

Answer: The inverse operation of taking the ninth root is raising the number to the power of 9.

Step-by-step solution

Understanding Inverse Operations

Inverse operations are the operations that can reverse the effect of the original operation. In other words, if the original operation takes you from point A to point B, then the inverse operation takes you back from point B to point A. Examples of inverse operations are addition and subtraction, multiplication and division, or square and square root.

Identifying the Given Operation

The given operation in this exercise is taking the ninth root. Mathematically, taking the ninth root of a number is the same as raising that number to the power of \(\frac{1}{9}\). For example, the ninth root of a number x can be written as \(x^{\frac{1}{9}}\).

Finding the Inverse Operation

Now, we need to find the inverse operation for taking the ninth root. Since taking the ninth root can be written as \(x^{\frac{1}{9}}\), the inverse operation should undo this effect and bring the result back to x. To do this, we need to raise the number to the power of 9, which is the reciprocal of \(\frac{1}{9}\). So, the inverse operation is exponentiation with the power of 9. In other words, the inverse operation for taking the ninth root is raising the given number to the power of 9. Mathematically, this can be written as \(x^{9}\).

Key concepts

Ninth Root

The ninth root of a number is a special type of root that reverses the process of raising a number to the ninth power. To find the ninth root of a number, you essentially look for a number that, when raised to the ninth power, gives you the original number. For example, if you have a number, such as 512, finding its ninth root means determining which number multiplied by itself nine times results in 512. Mathematically, this can be expressed as finding a number \( x \) such that \( x^9 = 512 \).
Using the concept of roots and exponents, taking the ninth root of \( x \) can be represented with the expression \( x^{\frac{1}{9}} \). This representation shows that you perform an inverse operation of raising to a power, which leads us to the next important concept, exponentiation, to understand the interplay between these operations.
Recognizing how roots correspond to fractional exponents helps in grasping more complex mathematical concepts with ease. By recognizing operations like taking the ninth root, you will be able to solve various equations and comprehend different problem-solving approaches more effectively.

Exponentiation

Exponentiation refers to the operation of raising a number, called the base, to a certain power, which is denoted as the exponent. This mathematical operation is fundamental in numerous areas of math and science, functioning as repeated multiplication.
When we write \( x^9 \), this implies multiplying the base \( x \) by itself eight more times (\( x \, \times \, x \, \times \, x \, \cdots \, x \), nine times total). Exponentiation extends beyond whole numbers, involving forms such as fractions and even negative numbers, thus providing a broad field of application.

• Whole-number exponents show simple repeated multiplication.
• Fractional exponents denote roots, such as \( x^{\frac{1}{9}} \), where the numerator indicates power and the denominator signifies the root.
• Negative exponents imply reciprocals, such that \( x^{-2} = \frac{1}{x^2} \).

When exploring inverse operations, exponentiation is crucial because it allows us to reverse processes like finding the ninth root by constructing equations that aid in problem-solving across various mathematical problems.

Mathematical Operations

Mathematical operations form the basis of computation and are utilized extensively in problem-solving to manipulate numbers and expressions. Among the essential operations are addition, subtraction, multiplication, and division, but the concepts of exponentiation and roots also play significant roles.
Let's delve into how these operations work as inverses to one another:

• Addition and subtraction are natural opposites, as subtraction undoes addition—subtracting a number from a sum recovers the original value.
• Multiplication and division are complementary, where division reverses multiplication, restoring the initial number by equally dividing it.
• Exponentiation and roots serve as each other's inverses. Raising a number to a power and then finding its corresponding root will return to the initial number, like raising \( x \) to the ninth power followed by taking the ninth root yields \( x \).

Using these operations fluently allows you to perform various algebraic processes, solve equations, and adjust expressions while understanding the intrinsic relationships between operations, especially through inverse procedures.