Problem 28 Convert to expressions with rati... [FREE SOLUTION]

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Calculus for Life Sciences

Marvin L. Bittinger, Neal Brand, John Quintanill

Biology

1 Edition

Chapter 1: Problem 28

Convert to expressions with rational exponents. \(\sqrt{x^{3}+4}\)

Short Answer

Expert verified

(x^{3}+4)^{1/2}

Step by step solution

Understand the Radical Expression

The given expression is \(\backslashsqrt{x^{3}+4}\). This is a square root, which can be expressed with a rational exponent.

Identify the Radical Form

The square root of a number \(n\) can be written as \((n^{1/2})\).

Apply the Rational Exponent

Using the property from Step 2, \(\backslashsqrt{x^{3}+4}\) becomes \((x^{3}+4)^{1/2}\).

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Key Concepts

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

Radical Expressions

A radical expression involves roots, like square roots or cube roots. The symbol for a square root is \(\backslashsqrt{}\), and for a cube root, \(\backslashsqrt[3]{}\). These symbols indicate we are taking the root of a number.
A radical expression can appear in various forms:

\( \backslashsqrt[3]{8} \)
\( \backslashsqrt{x+5} \)

To better understand and simplify radical expressions, it's helpful to represent them using rational exponents, too.

Square Root

The square root is a type of radical expression where the root is 2. The square root of a number \(x\) is written as \( \backslashsqrt{x} \).
This is equivalent to raising \(x\) to the power of \(1/2\):

\( \backslashsqrt{x} = x^{1/2} \)

In the original exercise, the expression \( \backslashsqrt{x^3 + 4} \) was given. We converted it to a rational exponent form by recognizing that a square root is the same as raising to the power of \(1/2\). Thus, \( \backslashsqrt{x^3 + 4} \) becomes \( (x^3 + 4)^{1/2} \).

Rational Exponent

A rational exponent is an exponent that is a fraction. For example, \(x^{1/2}\) and \(y^{3/4}\) are rational exponents. These can make it easier to handle and manipulate expressions involving roots.
When you see a radical, you can often rewrite it using a rational exponent:

The nth root of a number \(x\) is \( \backslashsqrt[n]{x} = x^{1/n} \).
Combining powers and roots can yield \( x^{m/n} \), which means \( ( \backslashsqrt[n]{x})^m \) or \( \backslashsqrt[n]{x^m} \).

Utilizing rational exponents simplifies many algebraic operations and can make solving problems much easier. Converting \( \backslashsqrt{x^3 + 4} \) to \( (x^3 + 4)^{1/2} \) is a perfect example of using a rational exponent.

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91影视

Convert to expressions with rational exponents. \(\sqrt{x^{3}+4}\)

Short Answer

Step by step solution

Understand the Radical Expression

Identify the Radical Form

Apply the Rational Exponent

Key Concepts

Radical Expressions

Square Root

Rational Exponent

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