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Chapter 1: Problem 3

(a) Using exponential notation, we can write \(\sqrt[3]{5}\) as (b) Using radicals, we can write \(5^{1 / 2}\) as (c) Is there a difference between \(\sqrt{5^{2}}\) and \((\sqrt{5})^{2}\) ? Explain.

Short Answer

Expert verified
(a) \(5^{1/3}\); (b) \(\sqrt{5}\); (c) No difference, both equal 5.

Step by step solution

01

Convert radical to exponential notation

The cubic root of 5, \( \sqrt[3]{5} \), can be expressed in exponential notation as \( 5^{1/3} \). In exponential terms, the expression \( a^{1/n} \) represents the nth root of \( a \).
02

Convert exponential to radical notation

The expression \( 5^{1/2} \) can be rewritten using radicals as \( \sqrt{5} \). In this context, \( a^{1/n} \) corresponds to \( \sqrt[n]{a} \), which is the nth root of \( a \).
03

Compare nested square root and squared square root

First, calculate \( \sqrt{5^2} \), which simplifies as follows:\[ \sqrt{5^2} = \sqrt{25} = 5 \]Next, calculate \( (\sqrt{5})^2 \), simplify it:\[ (\sqrt{5})^2 = 5^{1/2 \cdot 2} = 5^1 = 5 \]Comparing the two results, both \( \sqrt{5^2} \) and \( (\sqrt{5})^2 \) equal 5 and are therefore identical.

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Key Concepts

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

Understanding Radical Notation
Radical notation is a way of expressing roots of numbers using a special symbol called the radical sign, \( \sqrt{} \). This sign helps in identifying which root of a number is being referred to.
  • The expression under the radical sign is called the radicand.
  • The number or variable outside and on top of the sign is known as the index, which defines the degree of the root.
For instance, \( \sqrt[3]{5} \) represents the cubic root of 5. That means we are trying to find a number that, when multiplied by itself three times, results in 5. A radical can take any real number as a radicand, making it versatile and crucial in simplifying expressions and solving equations.
Decoding the Cubic Root
The cubic root, denoted as \( \sqrt[3]{} \) or \( x^{1/3} \), refers to a number that, when multiplied by itself three times, yields the original number. It's a specific type of root and is part of the broader category of nth roots.
  • For example, the cubic root of 8 is 2 because \( 2 \times 2 \times 2 = 8 \).
  • Cubic roots can apply to both positive and negative numbers, but remember, the result of a cubic root is always a real number.
In the exercise, \( \sqrt[3]{5} \) is written in exponential notation as \( 5^{1/3} \). This notation is particularly useful because it allows easier manipulation of expressions and equations involving roots, especially when working with powers.
The Simplicity of Square Root
The square root of a number, indicated by \( \sqrt{} \) or \( x^{1/2} \), is a value that, when multiplied by itself, gives the original number. It's one of the most fundamental mathematical operations and a cornerstone in understanding algebra.
  • The square root of 4 is 2 because \( 2 \times 2 = 4 \).
  • It is important to note that square roots usually operate within the realm of non-negative numbers.
In the given exercise, \( 5^{1/2} \) is transformed into \( \sqrt{5} \), highlighting the interchangeability of exponential and radical notations. Practically, this helps simplify problems and communicate solutions effectively when dealing with quadratic equations and geometrical computations.
Mastering Exponents
Exponents are a way of denoting how many times a number (the base) is multiplied by itself. Expressed as \( x^n \), it is an efficient shorthand vital in simplifying large multiplications.
  • For example, \( 3^4 \) means \( 3 \times 3 \times 3 \times 3 = 81 \).
  • In exponential notation, roots are expressed as fractional exponents, where the numerator represents the power, and the denominator represents the root.
In the exercise, we see exponents in different contexts: converting between \( \sqrt[3]{5} \) and \( 5^{1/3} \), as well as exploring the relationship between \( \sqrt{5^2} \) and \( (\sqrt{5})^2 \). Through this, exponents become indispensable in simplifying repeated multiplication and working in algebraic expressions seamlessly.

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Most popular questions from this chapter

Use a Special Factoring Formula to factor the expression. $$27 x^{3}+y^{3}$$

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$x^{-1 / 2}(x+1)^{1 / 2}+x^{1 / 2}(x+1)^{-1 / 2}$$

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Factor the expression completely. $$12 x^{3}+18 x$$

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