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Chapter 10: Problem 26

Use radical notation to write each expression. Simplify if possible. $$ (x-4)^{3 / 4} $$

Short Answer

Expert verified
\( \sqrt[4]{(x-4)^3} \) is the expression in radical form.

Step by step solution

01

Identify Radical Notation

Recognize that the expression \((x-4)^{3/4}\) involves a fractional exponent. The numerator (3) is the power, and the denominator (4) is the root. This means you can express it in radical notation as the fourth root of \((x-4)^3\).
02

Express Using the Radical Symbol

Rewrite the expression \((x-4)^{3/4}\) using radical notation: \[ \sqrt[4]{(x-4)^3} \]. This notation reflects taking the fourth root of \((x-4)\) raised to the third power.
03

Simplify the Expression (If Possible)

Consider if further simplification is possible. In this case, \( (x-4)^{3} \) inside the radical cannot be simplified further without additional information about \(x\). Therefore, this expression in radical form is \( \sqrt[4]{(x-4)^3} \).

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Key Concepts

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

Understanding Fractional Exponents
Fractional exponents can sometimes seem tricky, but they offer a powerful way to express complex operations using exponents. Fractional exponents signify both a power and a root.
  • The numerator of the fraction represents the power.
  • The denominator signifies the root.
For instance, in the expression \((x-4)^{3/4}\),the "3" in \(3/4\)is the power, and "4" is the root. This indicates you will raise \((x-4)\)to the third power and then take the fourth root.

By expressing numbers in this way, you can easily convert between exponential and radical forms, helping to simplify complex expressions.
Simplifying Radicals Made Easy
Simplifying radicals is an important skill when working with algebraic expressions, especially when they include fractional exponents.This process involves rewriting a radical expression in its simplest form. When you convert a fractional exponent into a radical expression, as shown in the solution for \((x-4)^{3/4}\),the outcome is \(\sqrt[4]{(x-4)^3}\).While you cannot simplify \((x-4)^{3}\) further without knowing more about "x," understanding the properties of radicals helps you simplify expressions when possible.

  • Identify common factors inside the radical and try to separate them.
  • Use the properties of roots to break down larger expressions.
This knowledge aids in recognizing when a radical can be simplified further or left as it is.
Solving Algebraic Expressions
Working with algebraic expressions like \((x-4)^{3/4}\)allows you to apply concepts of exponents and radicals in practical terms. These expressions frequently appear in various fields of mathematics and can represent numerous scenarios.Understanding how to translate between exponential and radical forms, as well as simplifying, is crucial when solving algebraic equations or expressions involving variables.Here are a few points to consider when dealing with algebraic expressions:
  • Identify the base and its exponents clearly.
  • Translate between different formats (fractional exponents and radicals).
  • Simplify when possible to make expressions easier to work with.
Mastering these concepts helps not only in simplifying expressions but also in graphing, solving equations, and understanding mathematical models.

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Most popular questions from this chapter

In electronics, the angular frequency of oscillations in a certain type of circuit is given by the expression \((L C)^{-1 / 2}\) Use radical notation to write this expression.

Use rational expressions to write as a single radical expression. $$ \frac{\sqrt[5]{b^{2}}}{\sqrt[10]{b^{3}}} $$

Use rational exponents to simplify each radical. Assume that all variables represent positive numbers. $$ \sqrt[12]{a^{8} b^{4}} $$

Identify the domain and then graph each function. $$f(x)=\sqrt{x-3}$$ use the following table. $$\begin{array}{|c|c|} \hline x & {f(x)} \\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 7 \\ \hline 12 & {} \\ \hline \end{array}$$

Write each integer as a product of two integers such that one of the factors is a perfect cube. For example, write 24 as \(8 \cdot 3,\) because 8 is a perfect cube. $$ 80 $$
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