Problem 19 - College Algebra

Question

Use the product rule to simplify the expressions in Exercises \(13-22\). In Exercises \(17-22,\) assume that variables represent nonnegative real numbers. $$\sqrt{x^{3}}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{x^{3}}\) is \(x^{\frac{3}{2}}\).

Step 1: Rewrite the square root as an exponent

First, let's acknowledge that the square root of a number is the same as raising that number to the power of \( \frac{1}{2}\). So, the expression \(\sqrt{x^{3}}\) can be rewritten as \(x^{\frac{3}{2}}\).

Step 2: Simplify the exponent

Having \(x^{\frac{3}{2}}\) means that we have \(x\) raised to \(\(\frac{3}{2}\)\). This is already simplified, and doesn't need any further adjustment since \(x\) is a nonnegative real number.

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in algebra that describe how many times a number, known as the base, is multiplied by itself. In mathematical terms, if we have a number "a" raised to the power "n," it is expressed as \(a^n\). Here, "n" is the exponent.

• If the exponent is a whole number, it indicates repeated multiplication.
• If the exponent is a fraction, as in \(x^{\frac{3}{2}}\), it represents a combination of root and power operations.

For instance, \(x^{\frac{3}{2}}\) can be understood as the square root of \(x^3\) or equally as \((\sqrt{x})^3\). It combines both multiplying \(x\) with itself three times and then taking the square root of the result, or alternatively treating it as the square root of \(x\), taken three times. This duality makes fractional exponents very powerful and versatile in simplifying complex expressions.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their most basic form without changing their value. This involves minimizing the complexity of the expression so it becomes easier to understand or work with further.

• Start by rewriting complex roots and fractional exponents in a more uniform manner.
• Look for opportunities to apply rules of arithmetic, particularly regarding exponents, such as the product and quotient rules.

In our example of \(\sqrt{x^3}\), it was simplified to \(x^{\frac{3}{2}}\) by converting the square root into an exponent which clearly demonstrates power.
The simplification step often reveals insights into the relationship between the elements of the expression—especially when visualizing how the elements interact within the equation. Removing unnecessary complexity by streamlining expressions is key for creating efficient solutions and establishing clarity.

Importance of Nonnegative Real Numbers

The clarification that variables represent nonnegative real numbers is crucial for avoiding complications in the operations involving radicals and fractional exponents.

• A nonnegative number is one that is either positive or zero.
• Real numbers are those that can be found on the number line, encompassing both rational and irrational numbers.

When working with expressions like \(\sqrt{x^3}\), defined over nonnegative real numbers, we ensure that the roots are well-defined and the results remain within the realm of real numbers without diving into complex numbers, which can occur with negative inputs.
This consideration simplifies both solving and understanding problems by keeping all square roots and fractional exponents valid within the real number system, thus preventing undefined or complex results.