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Chapter 1: Problem 35

Convert to radical notation. \(b^{-1 / 3}\)

Short Answer

Expert verified
\(\frac{1}{\sqrt[3]{b}}\)

Step by step solution

01

- Understand the Exponential Form

The given expression is in exponential form: \(b^{-1 / 3}\). The exponent \(-1 / 3\) can be split into two parts: \(-1\) and \(1/3\).
02

- Apply the Negative Exponent Rule

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). Applying this rule, we get \(b^{-1 / 3} = \frac{1}{b^{1 / 3}}\).
03

- Convert to Radical Notation

The expression \(b^{1 / 3}\) represents the cube root of \(b\). Hence, \(b^{1 / 3} = \sqrt[3]{b}\). Therefore, \(\frac{1}{b^{1 / 3}} = \frac{1}{\sqrt[3]{b}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a way to express repeated multiplication of a number by itself. If a number is raised to an exponent, this means you multiply the number by itself as many times as the exponent indicates. For instance, in the expression 2^3, the base number is 2, and the exponent is 3. This means you multiply 2 by itself three times: 2 * 2 * 2. In general, the expression a^n means you multiply 'a' by itself 'n' times.
Negative Exponent Rule
The negative exponent rule helps to simplify expressions with negative exponents. According to this rule, any number raised to a negative exponent can be converted to its reciprocal with a positive exponent. Mathematically, this rule is represented as: \(a^{-n} = \frac{1}{a^n}\). This means that, for example, 3^{-2} is the same as \(\frac{1}{3^2} = \frac{1}{9}\). This rule is very useful for converting expressions and solving equations that involve negative exponents.
It is important to remember that this rule applies universally to all numbers and helps in simplifying complex problems.
Cube Root
In mathematics, the cube root of a number is a special value that, when used in a multiplication three times, gives that number. The cube root is the inverse operation of cubing a number. For instance, the cube root of 27 is 3 because 3 * 3 * 3 = 27. The cube root is denoted by the radical symbol with a little three above the n 鈭�, so 鈭歔3]{y} denotes the cube root of y.
In standard form, if b is any number, then \(b^{1/3} = \sqrt[3]{b}\). This notation is frequently used in algebra to solve equations involving cube roots and to simplify expressions.

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Most popular questions from this chapter

A key \(n\) notes (including half-notes). above the A above middle C generates a sound wave proportional to $$ y=\sin \left(880 \cdot 2^{n / 12} \pi t\right) $$ where \(t\) is in seconds. Use this equation to solve the following exercises. Find the note with a frequency of \(4 \cdot 440=1760 \mathrm{~Hz}\)

a) Show that \(\cos 2 t=\cos ^{2} t-\sin ^{2} t\) b) Show that \(\cos 2 t=2 \cos ^{2} t-1\). (Hint: Use part (a) and a Pythagorean Identity.)

Use a calculator to evaluate the following trigonometric functions. $$ \cos 125^{\circ} $$

Find all solutions of the given equation. $$ \sin ^{2} t-2 \sin t-3=0 $$

Use a calculator to find the values of the following trigonometric functions. \(\tan 5^{\circ}\)
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