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Chapter 10: Problem 3

Use radical notation to rewrite each expression. Simplify if possible. $$ 27^{1 / 3} $$

Short Answer

Expert verified
The expression in radical form is \(\sqrt[3]{27}\), which simplifies to 3.

Step by step solution

01

Understand the Exponent

The expression given is \(27^{1/3}\). The exponent \(1/3\) indicates that we need to take the cube root of 27. When you see a fractional exponent like this, the numerator represents a power, and the denominator represents a root. Here, since the numerator is 1, we are only concerned with the cube root.
02

Rewrite in Radical Notation

Using the rule for fractional exponents, \(a^{m/n} = \sqrt[n]{a^m}\), we rewrite \(27^{1/3}\) as \(\sqrt[3]{27}\). This means we're looking for a number that, when multiplied by itself three times, results in 27.
03

Simplify the Radical

Think about what number cubed equals 27. We know that \(3 \times 3 \times 3 = 27\). Therefore, the cube root of 27 is 3. This means \(\sqrt[3]{27} = 3\).

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Key Concepts

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

Fractional Exponents
Fractional exponents, like the one seen in the expression \(27^{1/3}\), are not as complicated as they might seem. They provide a way to express roots using powers. In a fractional exponent \(a^{m/n}\), the numerator \(m\) acts as a power, and the denominator \(n\) indicates the root. This means that \(a^{1/3}\) translates to the cube root of \(a\). Using fractional exponents is beneficial because it allows us to switch easily between powers and roots, offering a concise way to represent calculations.
  • The base number here is 27.
  • The exponent is \(1/3\), meaning a cube root since 1 is the power and 3 is the root.
It is crucial to remember that fractional exponents not only provide an alternative to radical notation but also simplify algebraic operations by abiding to exponent rules.
Cube Root
The cube root is an important mathematical operation that focuses on finding a number which, when multiplied by itself three times, gives the original number. In radical notation, this is represented as \(\sqrt[3]{a}\), where \(a\) is the number you're taking the cube root of.
To solve \(\sqrt[3]{27}\), you need to determine what number cubed equals 27. We know that 3 cubed (\(3 \times 3 \times 3\)) equals 27. Therefore, the cube root of 27 is simply 3.
  • \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
  • Cube roots are the inverse operation of cubing a number.
Understanding cube roots helps in solving equations and simplifying expressions involving radical notation and fractional exponents.
Simplifying Radicals
Simplifying radicals involves reducing the radical expression to its simplest form. In the case of \(\sqrt[3]{27}\), it's already given in a form that can be simplified easily. Simplifying requires finding the cube root, which as we found, is 3.
Here are quick steps you might follow for simplifying cube roots:
  • Determine if the number is a perfect cube (like 27).
  • If the number is a perfect cube, find the integer that, when cubed, results in the number under the radical.
In our case, 27 is a perfect cube as \(3^3 = 27\). Thus, \(\sqrt[3]{27} = 3\).
Simplifying radicals is crucial for reducing algebraic expressions into their simplest forms, helping in understanding and solving problems involving radicals more effectively.

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Most popular questions from this chapter

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x}}\)

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{\sqrt{x}}{\sqrt{x}+\sqrt{y}}\)

Solve. Give exact answers and two-decimal-place approximations where appropriate. The tallest structure in the United States is a TV tower in Blanchard, North Dakota. Its height is 2063 feet. A 2382 -foot length of wire is to be used as a guy wire attached to the top of the tower. Approximate to the nearest foot how far from the base of the tower the guy wire must be anchored. (Source: U.S. Geological Survey)

The cost \(C(x)\) in dollars per day to operate a small delivery service is given by \(C(x)=80 \sqrt[3]{x}+500,\) where \(x\) is the number of deliveries per day. In July, the manager decides that it is necessary to keep delivery costs below \(\$ 1620.00 .\) Find the greatest number of deliveries this company can make per day and still keep overhead below \(\$ 1620.00\)

In psychology, it has been suggested that the number S of nonsense syllables that a person can repeat consecutively depends on his or her IQ score I according to the equation \(S=2 \sqrt{I}-9\). Use this relationship to estimate the IQ of a person who can repeat 15 nonsense syllables consecutively.
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