Unit 6 radical functions homework 1 nth roots simplifying radicals – Delving into the intricacies of unit 6 radical functions homework 1, this exploration unravels the concepts of nth roots and radical simplification. Through a comprehensive examination of definitions, examples, and applications, we embark on a journey to master these fundamental mathematical operations.
By exploring the relationship between nth roots and exponents, we lay the foundation for understanding the intricacies of radical expressions. We delve into the practical steps involved in simplifying radicals, including rationalizing the denominator, and discover the power of conjugate pairs in streamlining these calculations.
1. Nth Roots: Unit 6 Radical Functions Homework 1 Nth Roots Simplifying Radicals
Nth roots are a fundamental concept in mathematics that involves finding a number that, when raised to a specified power, equals a given number. The nth root of a number a, denoted as √[n]a, is a number b such that b^n = a.
Here, n is a positive integer called the index or order of the root.
For example, the square root of 9 is 3, written as √[2]9 = 3, because 3^2 = 9. Similarly, the cube root of 27 is 3, written as √[3]27 = 3, because 3^3 = 27.
Nth roots have a close relationship with exponents. The nth root of a number a is equivalent to raising a to the power of 1/n, that is, √[n]a = a^(1/n). This relationship allows us to convert between nth roots and exponents.
2. Simplifying Radicals
Simplifying radicals involves transforming a radical expression into an equivalent form that is easier to understand and work with. The goal is to remove any irrationality from the denominator of the radical and to express the radical in its simplest form.
To simplify a radical, we can use the following steps:
- Factor the radicand into a product of prime factors.
- Identify any perfect squares in the radicand and take them outside the radical.
- Rationalize the denominator by multiplying and dividing by the conjugate of the denominator.
For example, to simplify √[2](12/5), we first factor the radicand: 12/5 = 2^2 – 3/ 5. We then take the perfect square 2^2 outside the radical, which gives us 2√[2](3/5). Finally, we rationalize the denominator by multiplying and dividing by √[2](5): 2√[2](3/5) – √[2](5)/√[2](5) = 6√[2]/5.
3. Radical Functions
Radical functions are functions that involve radicals in their expression. They are defined as functions of the form f(x) = √[n]x, where n is a positive integer and x is the independent variable.
The domain of a radical function is the set of all non-negative real numbers, since the radicand must be non-negative for the function to be defined. The range of a radical function depends on the index n.
Radical functions have several important properties, including:
- They are continuous on their domain.
- They are differentiable on their domain except at x = 0.
- They are increasing on their domain.
4. Homework 1
The following table lists the problems in Homework 1 and provides step-by-step solutions for each problem.
| Problem | Solution |
|---|---|
| Simplify √[3](27) | √[3](27) = 3 |
| Simplify √[4](81) | √[4](81) = 3 |
| Simplify √[5](32) | √[5](32) = 2 |
Q&A
What is the definition of an nth root?
An nth root of a number a is a number b such that b^n = a.
How do you simplify radicals?
To simplify a radical, factor out any perfect squares from the radicand and then rationalize the denominator if necessary.
What are conjugate pairs?
Conjugate pairs are two expressions that differ only by a sign, such as (a + b) and (a – b).
