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Chapter 1: Problem 17

Convert to expressions with rational exponents. \(\sqrt{x^{3}}\)

Short Answer

Expert verified
The expression \(\root{x^{3}}\) can be written as \(x^{3/2}\).

Step by step solution

01

Identify the Expression

The given expression is \(\root{x^{3}}\). The goal is to convert this radical expression to an expression with a rational exponent.
02

Understand the Radical Notation

A radical expression \(\root[n]{a}\) can be written as \(a^{1/n}\). In this case, \(a = x^{3}\) and the index of the root \((n)\) is 2, since it is a square root.
03

Apply the Rational Exponent Rule

Using the rule mentioned in the previous step, \(\root{2}{x^{3}}\) can be written as \((x^{3})^{1/2}\).
04

Simplify the Expression

Apply the power rule of exponents to simplify \((x^{3})^{1/2}\). When you raise a power to a power, you multiply the exponents: \(x^{3 \times 1/2} = x^{3/2}\).

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Key Concepts

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

Radical Notation
Radical notation is a way to represent roots of numbers using the radical symbol (鈭�). For example, the square root of a number 'a' is written as \(\root{2}{a}\). In general, the nth root of a number 'a' is written as \(\root{n}{a}\). This notation shows that we are looking for a number which, when raised to the power of n, gives us the number 'a'.

In our exercise, we have \(\root{x^{3}}\), which means the square root of \(x^{3}\). The number inside the radical sign is called the radicand, and the small number outside, which is rarely shown for square roots but is still considered to be 2, is the index or the degree of the root.
Exponent Rules
Exponent rules help us to simplify expressions involving powers of numbers or variables. Here are some important rules to remember:

1. **Product Rule**: When multiplying like bases, you add their exponents. For example: \(a^{m} \times a^{n} = a^{m+n}\).
2. **Quotient Rule**: When dividing like bases, you subtract the exponents. For example: \(a^{m} / a^{n} = a^{m-n}\).
3. **Power Rule**: To raise a power to a power, you multiply the exponents. For instance, \( (a^{m})^{n} = a^{m \times n}\).

In the given exercise, the expression \( (x^{3})^{1/2} \) uses the power rule. We multiply the exponents (3 and 1/2) together to simplify: \( (x^{3})^{1/2} = x^{3 \times 1/2} = x^{3/2} \).
Power Rule
The power rule is one of the most commonly used exponent rules. According to this rule, when you raise an expression that already has an exponent to another power, you multiply the exponents.

For example:

\((a^{m})^{n} = a^{m \times n}\)

Let's see how we applied this rule in our exercise:
If you start with \(x^{3}\) and then raise it to the 1/2 power, you multiply the existing exponent (3) by the new exponent (1/2):

\((x^{3})^{1/2}\)

Using the power rule, this simplifies to:

\((x^{3})^{1/2} = x^{3 \times 1/2} = x^{3/2}\)

This result shows x raised to the 3/2 power, which is another way of expressing the square root of \(x^{3}\).

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Most popular questions from this chapter

Use a calculator to find the degree measure of an acute angle whose trigonometric function is given. \(\tan t=0.84\)

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Sound Waves. The pitch of a sound wave is measured by its frequency. Humans can hear sounds in the range from 20 to \(20,000 \mathrm{~Hz}\), while dogs can hear sounds as high as \(40,000 \mathrm{~Hz}\). The loudness of the sound is determined by the amplitude. \({ }^{22}\). The note \(A\) below middle \(C\) on a piano generates a sound modeled by the function \(g(t)=4 \sin (440 \pi t)\), where \(t\) is in seconds. Find the frequency of \(A\) below middle \(C\).
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