Problem 19 - College Algebra Essentials

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

\(x^{\frac{3}{2}}\)

Step 1: Recognizing the square root

In given expression \(\sqrt{x^{3}}\), this is a square root function, which basically means we're finding the number that, when multiplied by itself, will result in \(x^{3}\).

Step 2: Simplifying the square root

We can rewrite the square root of \(x^{3}\) as Cube root of \(x\) Squared, because square root of \(x^{3}\) can be written as \((x^{\frac{1}{2}})^{3}\) and using power of power rule, we can write it as \((x^{\frac{3}{2}})\).

Step 3: Expressing the final answer

So, \(\sqrt{x^{3}}\) simplifies to \((x^{\frac{3}{2}})\), for x is nonnegative real Number.

Key Concepts

Simplifying Square Roots

When you encounter an expression like \(\sqrt{x^3}\), you are dealing with a square root. The goal is to simplify the expression to its simplest form. To do this, we can utilize the property of square roots which allows us to express the square root in terms of fractional exponents.

• Recognize that \(\sqrt{x^3}\) is the same as \((x^3)^{1/2}\). This is because taking the square root of a number is equivalent to raising it to the power of \(1/2\).
• To further simplify, apply the property of exponents: \((a^m)^n = a^{m \cdot n}\).
• In this instance, (\(x^3)^{1/2}\) simplifies to \(x^{3 \cdot 1/2} = x^{3/2}\).

These steps showcase how you can convert complex square roots into a fraction of powers, making the expression easier to handle or use in subsequent calculations.

Exponents and Powers

Exponents and powers are fundamental concepts in algebra. They indicate how many times you multiply a number by itself. When you are working with expressions that have exponents, such as \(x^3\) or \(x^{3/2}\), understanding how to manipulate these using rules is key.

The Power Rule for Exponents:

• To handle expressions like \((a^m)^n\), use the power rule: \((a^m)^n = a^{m\cdot n}\).
• This rule allows you to multiply the exponents together when raising an exponential expression to another power.

Fractional Exponents:

• Fractional exponents, like \(x^{1/2}\), indicate roots. For example, \(x^{1/2}\) represents the square root of \(x\).
• A key advantage of fractional exponents is that they easily represent roots and make multiplication and division simpler.

By mastering these rules, you can simplify expressions and perform calculations with greater efficiency.

Nonnegative Real Numbers

In algebra, dealing with nonnegative real numbers often simplifies problems. These are numbers that are zero or positive, which is crucial when you are using functions like square roots. The concept becomes particularly important in expressions like \(\sqrt{x^3}\) because a square root technically requires the input to be nonnegative to avoid complex numbers.

• Nonnegative numbers ensure that operations like square rooting result in real numbers.
• By limiting variables to nonnegative real numbers in problems, we prevent undefined or imaginary results.
• It allows for simplifications, such as translating a square root to a fractional exponent form without complications.

Thus, working with nonnegative real numbers is essential in ensuring that mathematical operations remain within the realm of real numbers, streamlining the simplification and manipulation of expressions.