Problem 532 - Elementary Algebra

Question

In the following exercises, write with a rational exponent. (a) \(\sqrt[3]{7 c}\) (b) \(\sqrt[7]{12 d}\) (c) \(3 \sqrt[4]{5 f}\)

Step-by-Step Solution

Answer

(a) (7c)^{1/3} (b) (12d)^{1/7} (c) 3(5f)^{1/4}

Step 1 - Understand the problem

The problem asks to convert each given expression, which is in radical form, to an expression with a rational exponent.

Step 2 - Write the radical in exponent form

Recall that the radical form \(\root{n}{x}\) can be written as \({x}^{1/n}\).

Step 3 - Convert each given radical expression

(a) For \(\root{3}{7c}\): Use the rule to write it as \((7c)^{1/3}\).

Step 4 - Convert the second expression

(b) For \(\root{7}{12d}\): Use the rule to write it as \((12d)^{1/7}\).

Step 5 - Convert the third expression

(c) For \((3 \root{4}{5f})\): Factor out the constant and then apply the rule to \(\root{4}{5f}\) to write it as \((3 \times (5f)^{1/4})\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. They are denoted by the radical symbol (√) or in a more general form like \(\root{n}{x}\). This notation represents the nth root of a number or expression.
For example, \(\root{3}{7c}\) means the cube root of \(\textbf{7c}\). To convert into rational exponents, we use the principle that \(\root{n}{x} = x^{1/n}\). This way, we can express roots as exponents, simplifying calculations and algebraic manipulations.

Exponent Rules

Understanding exponent rules is crucial in algebra. Exponents denote repeated multiplication and are written as \(x^a\), where the base is \(\textbf{x}\) and the exponent is \(\textbf{a}\). Here are a few fundamental rules:

• \textbf{Product Rule:} \(x^a \times x^b = x^{a+b}\)
• \textbf{Quotient Rule:} \(x^a / x^b = x^{a-b}\)
• \textbf{Power Rule:} \((x^a)^b = x^{a \times b}\)

When dealing with rational exponents, the form changes but these rules still apply. Converting a radical to a rational exponent allows the use of these exponent rules for easier manipulation.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and operators (such as +, -, *, /). Expressions like \(\textbf{7c}\)blend constants (7) with variables (c). In radical form, we encounter expressions like \(\root{3}{7c}\). Converting these to rational exponents simplifies them, making manipulation straightforward.
For instance, \(\root{3}{7c}\) becomes \( (7c)^{1/3} \). This way, we can apply exponent rules to perform further algebraic operations or simplify the expression efficiently.

Mathematical Notation

Mathematical notation is the system used to write mathematical concepts clearly and concisely. When learning about exponents and radicals, knowing the correct notation is essential.
Radicals use the symbol √ or the notation \(\(\root{n}{x}\)\). Exponents are written in the form \(x^a\). To convert between these forms, remember the key relationship: \(\root{n}{x} = x^{1/n}\). Using this notation ensures clarity and accuracy in expressing mathematical ideas.
Always use proper notation to avoid confusion and to make the problems easier to understand and solve.