Problem 25 - Intermediate Algebra

Question

Use radical notation to write each expression. Simplify if possible. $$ (7 x+2)^{2 / 3} $$

Step-by-Step Solution

Answer

The radical notation is \(\sqrt[3]{(7x + 2)^2}\).

Step 1: Understanding the Radical Notation

Radical notation involves writing expressions with roots. The expression \((7x + 2)^{2/3}\) can be re-written using a radical. Specifically, any expression in the form of \(a^{m/n}\) can be converted to \(\sqrt[n]{a^m}\).

Step 2: Convert to Radical Notation

Apply the rule from Step 1. The expression \((7x + 2)^{2/3}\) converts to radical notation as \(\sqrt[3]{(7x + 2)^2}\). This means we take the cube root (since the denominator of the exponent is 3) of \((7x + 2)\) squared.

Step 3: Simplify if Necessary

In this case, \(\sqrt[3]{(7x + 2)^2}\) is already in its simplest form, as there are no further algebraic simplifications to be made.

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. The notation for an exponent involves two parts: the base and the exponent. For example, in \(a^b\), \(a\) is the base, and \(b\) is the exponent. This means that \(a\) is multiplied by itself \(b\) times.
There are some special rules that govern the use of exponents:

• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Power Rule: \((a^m)^n = a^{m \times n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) where \(a eq 0\)
• Zero Exponent Rule: \(a^0 = 1\) where \(a eq 0\)
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\) where \(a eq 0\)

Exponents are especially useful in radical notation because they allow expressions to be written in an alternate form, making them easier to manipulate and understand.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form without changing their value. This process usually makes mathematical expressions easier to work with. When simplifying expressions involving exponents, apply the rules of exponents wherever possible.
For example, consider the expression \((7x + 2)^{2/3}\). By converting to radical notation, it becomes \(\sqrt[3]{(7x + 2)^2}\). Here, "simplify" means to ensure there's no further simplification possible.

• Factor where possible to reduce the base expression.
• If any common factors exist within the base, simplify them further.
• Make sure there are no further operations within the radical or exponential expression that can simplify the expression.

By tending to these aspects, you ensure the expression is at its simplest possible form.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and arithmetic operations. They can represent a wide variety of mathematical problems and solutions. In the given expression, \((7x+2)^{2/3}\), it's important to recognize the "\(7x+2\)" as an algebraic term.
Algebraic expressions like this can be simplified or transformed using operations on exponents, radicals, and other algebraic properties. Understanding the basic components of algebraic expressions can help you manipulate them more effectively:

• Terms: The parts of an expression separated by plus or minus signs. In "\(7x+2\)", both "\(7x\)" and "\(2\)" are terms.
• Coefficients: The numerical part of terms. In "\(7x\)", 7 is the coefficient.
• Variables: Represent unknown values and can change, in this case, \(x\).
• Constants: Numbers on their own, such as "2" in the expression above.

By practicing with these elements, you can gain confidence in manipulating algebraic expressions across different contexts.