Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 53-56: Rewrite the expression using a radical. (a) \(8-y^{1 / 3}\) (b) \((8-y)^{1 / 3}\)

Step-by-Step Solution

Answer

(a) \(8-\sqrt[3]{y}\), (b) \(\sqrt[3]{8-y}\).

Step 1: Identify the Expression

We need to rewrite each given expression using radical notation. We are given two expressions: (a) \(8-y^{1/3}\) and (b) \((8-y)^{1/3}\).

Step 2: Rewrite Expression (a)

The term \(y^{1/3}\) represents the cube root of \(y\). Using radical notation, we write \(y^{1/3}\) as \(\sqrt[3]{y}\). Therefore, the expression \(8-y^{1/3}\) becomes \(8 - \sqrt[3]{y}\).

Step 3: Rewrite Expression (b)

The term \((8-y)^{1/3}\) represents the cube root of the entire expression \(8-y\). Using radical notation, we write \((8-y)^{1/3}\) as \(\sqrt[3]{8-y}\). Thus, the expression is now \(\sqrt[3]{8-y}\).

Key Concepts

Cube Root

The cube root is a fundamental concept in mathematics, especially when dealing with multiplying numbers by themselves multiple times. When we say cube root, it means finding a number that, when multiplied by itself three times, equals the original number. For example, the cube root of 8 is 2 because 2 multiplied by 2 three times is 8 (\(2 \times 2 \times 2 = 8\)).In mathematical notation, the cube root of a number is written using radical notation with a small three over the radical sign, like this: \(\sqrt[3]{x}\).

• The cube root symbol indicates the operation of finding which number, when cubed (raised to the power of three), yields the number inside the symbol.
• In the given expression, for instance, \(y^{1/3}\) translates directly into \(\sqrt[3]{y}\).

Remember, recognizing cubes and cube roots is essential in solving various mathematical problems and simplifying expressions.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a specific value or computation. They are the language of math, allowing us to communicate more complex ideas succinctly. Expressions can vary from simple, like single numbers or variables, to more complex arrangements involving operations such as addition, subtraction, multiplication, division, and even the use of roots and exponents.

• An example of a mathematical expression is \(8 - y^{1/3}\), which includes constants, variables, and an exponent.
• Expressions can be rewritten or simplified using various mathematical rules, like turning \(y^{1/3}\) into \(\sqrt[3]{y}\), making it easier to understand or solve.
• Knowing how to manipulate and understand expressions is crucial in all areas of mathematics, from basic arithmetic to advanced calculus.

Exponents and Radicals

In mathematical operations, exponents and radicals are tools that help us express and manipulate numbers in different forms.

• An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, \(y^{1/3}\) uses the exponent \(1/3\).
• This specific exponent indicates the root: the cube root in this case. It demonstrates the fractional nature and involves breaking numbers down rather than building them up.
• Radicals provide an alternative way to symbolize roots, making problems easier to read and solutions easier to find. For example, the expression \(y^{1/3}\) translates into the radical form of \(\sqrt[3]{y}\).

Understanding how to convert exponents to radicals and vice versa is vital for simplifying expressions and solving equations efficiently.