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Chapter 10: Problem 23

Use radical notation to write each expression. Simplify if possible. $$ (2 x)^{3 / 5} $$

Short Answer

Expert verified
The simplified radical expression is \( \sqrt[5]{8x^3} \).

Step by step solution

01

Understand the Problem

The expression given is \((2x)^{3/5}\). Our task is to express this using radical notation and simplify it if possible.
02

Convert to Radical Form

A fractional exponent, \(a^{m/n}\), can be rewritten in radical form as \(\sqrt[n]{a^m}\). Therefore, \((2x)^{3/5}\) can be expressed as \(\sqrt[5]{(2x)^3}\).
03

Simplify Inside the Radical

Now, simplify \((2x)^3\) inside the radical. We calculate \((2x)^3 = 2^3 \cdot x^3\), which simplifies to \(8x^3\).
04

Express the Final Simplified Radical

Substitute the simplified form back into the radical expression: \(\sqrt[5]{8x^3}\). This is the expression in simplified radical notation.

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Key Concepts

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

Fractional Exponents
Fractional exponents are a way of representing powers and roots together in one expression. They are an essential concept in algebra, offering a more compact way to handle mathematical operations involving roots and powers. In general, a fractional exponent like \(a^{m/n}\) signifies two operations: raising \(a\) to the power of \(m\), and then taking the \(n\)-th root of the result.
For example:
  • \(a^{1/2}\) represents the square root of \(a\), i.e., \(\sqrt{a}\).
  • \(a^{1/3}\) denotes the cube root, or \(\sqrt[3]{a}\).
  • \(a^{3/2}\) would first square \(a\) and then take the square root of the result, i.e., \(\sqrt{a^3}\).
This notation is incredibly useful when simplifying complex expressions, as it enables easy manipulation of powers and roots together.
Simplifying Radicals
Simplifying radicals involves rewriting a radical expression in its simplest form. This often means breaking down the expression inside the radical to reduce it to the simplest terms possible.
To simplify a radical, such as \(\sqrt[n]{a^m}\), follow these steps:
  • Check if the expression inside the radical can be factored into smaller powers matching the root. For instance, \((2x)^3\) simplifies by calculating \((2x)^3 = 2^3 \cdot x^3 = 8x^3\).
  • Once simplified, ensure there are no perfect powers within the radical that can be further reduced.
By simplifying radicals, you make the expression easier to understand and work with, eliminating unnecessary complexity.
Algebraic Expressions
Algebraic expressions form the foundation of algebra, representing numbers, operations, and variables together in a coherent form. Understanding and manipulating these expressions is crucial for solving equations and problems in algebra.
An algebraic expression like \((2x)^{3/5}\) includes:
  • A base, here \(2x\), composed of a number and a variable being acted upon.
  • Exponents, which tell you how many times to multiply the base by itself.
Mastering algebraic expressions involves learning to recognize and apply rules of arithmetic, such as multiplication, division, distribution, and powers, seamlessly. This allows you to transform and simplify expressions for easier calculation or further algebraic manipulation, as seen in converting \((2x)^{3/5}\) into \(\sqrt[5]{8x^3}\).

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Most popular questions from this chapter

Simplify each exponential expression. $$\left(-14 a^{5} b c^{2}\right)\left(2 a b c^{4}\right)$$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[6]{4} $$

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9.2,\) because 9 is a perfect square. $$ 45 $$

Fill in each box with the correct expression $$ \square \cdot a^{2 / 3}=a^{3 / 3}, \text { or } a $$

Factor the common factor from the given expression. $$ x^{-1 / 3} ; 5 x^{-1 / 3}+x^{2 / 3} $$
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