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Chapter 7: Problem 37

Rational Exponents Write an equivalent expression using exponential notation. $$\sqrt{30}$$

Short Answer

Expert verified
30^{1/2}

Step by step solution

01

Understand the Radical Notation

Identify the given expression. For this problem, the given expression is \(\text{\( \sqrt{30} \)}\). This is a square root.
02

Convert the Square Root to a Rational Exponent

Recall that a square root can be expressed as an exponent of \(1/2\). Therefore, \(\text{\( \sqrt{30} \)}\) can be written as \(\text{\( 30^{1/2} \)}\).

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Key Concepts

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

Square Root
The square root is one of the most common mathematical operations.
It's the process of determining which number, when multiplied by itself, yields the original number.
In mathematical terms, if \(\sqrt{x} = y\), then \(y \cdot y = x\) or \( y^2 = x\).
For example, \(\sqrt{25} = 5\) because \(5^2 = 25\).
The symbol \(\sqrt{ } \) is used to denote the square root operation, often referred to as the *radical symbol*.
In math problems, you'll frequently encounter square roots as they simplify numbers in expressions and equations.
Exponential Notation
Exponential notation is a way to represent numbers as exponents.
It is especially useful for expressing very large or very small numbers compactly.
For example, \(10^3\) means \(10 \cdot 10 \cdot 10 = 1000\).
In our exercise, we saw that the square root of 30 can be written in exponential form as \(30^{1/2}\).
This is because, in general, the square root of a number \(a\) can be written as \(a^{1/2}\).
Radical Notation
Radical notation represents roots of numbers.
The most common radical operation is the square root, represented as \(\sqrt{a}\).
Other roots include the cube root \(\sqrt[3]{a}\) and the fourth root \(\sqrt[4]{a}\), and so on.
To convert from radical notation to exponential notation, use the rule:
For any number \( a \) and root \( n \), \( \sqrt[n]{a} = a^{1/n} \).
For example, \( \sqrt[3]{8} = 8^{1/3} = 2\), because \(2^3 = 8\).
Understanding radical notation is key to mastering exponents and roots.

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. (See the graph above.) Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}}\) Find the absolute value of each complex number. $$|8-6 i|$$

Simplify. $$\left(\frac{1}{2}-\frac{1}{3} i\right)^{2}-\left(\frac{1}{2}+\frac{1}{3} i\right)^{2}$$

Find the midpoint of the segment with the given endpoints. $$ (-1,2) \text { and }(1,-3) $$

Factor completely. $$2 x-63+x^{2}$$

Classify each of the following as either an arithmetic sequence, a geometric sequence, an arithmetic series, a geometric series, or none of these. \(3,6,12,24, \dots\)
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