Problem 19 - Algebra and Trigonometry

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

Step 1: Rewrite the Square Root

The square root of a cubed number \(x^{3}\) can be rewritten as the cubed root of \(x\) squared, \((\sqrt[3]{x})^{2}\).

Step 2: Apply the Product Rule

Using the product rule, we can break this down further into the product of the cubed root of \(x\) and the cubed root of \(x\), or \( \sqrt[3]{x} * \sqrt[3]{x}\).

Step 3: Combine to Simplify

Combining these terms provides a simplified expression of \(x^{2/3}\).

Key Concepts

Square Root

The square root is a mathematical operation that finds the original number which, when multiplied by itself, gives the specified value. For example, the square root of 9 is 3 because 3 multiplied by itself gives 9.
When dealing with variables and exponents, the square root can be expressed in exponential form. For instance, the square root of a number like \(x\), is denoted as \(x^{1/2}\). This notation helps when performing operations with other exponents as it allows for consistent application of the rules of exponents.

• For a term like \(\sqrt{x^3}\), rewriting it as \((x^3)^{1/2}\) helps simplify the expression.
• By applying the powers, this becomes \(x^{3/2}\).

Understanding square roots in this way is crucial for algebraic manipulation and simplification of terms.

Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 * 3 * 3 equals 27.
With cube roots, we use an exponent of \(1/3\) to represent the operation. So, the cube root of a variable \(x\) is expressed as \(x^{1/3}\). This form is particularly useful when dealing with expressions that involve other exponents.

• When simplifying an expression like \(\sqrt{(x^3)}\), thinking of it in terms of cube roots can help clarify each step.
• Rewriting \(\sqrt{x^3}\) as \((\sqrt[3]{x})^2\) uses the concept of breaking it down into smaller, more manageable parts.

This understanding aids in simplifying and manipulating algebraic expressions effectively.

Exponents

Exponents are powers that indicate how many times a number is multiplied by itself. For example, \(x^2\) means \(x\) is multiplied by \(x\) once, resulting in \(x*x\).
When we have complex terms like \(x^{3/2}\) or \(x^{1/3}\), these represent fractional exponents. Fractional exponents are a way to express roots and powers together. The numerator signifies the power and the denominator the root.

• \(x^{3/2}\) is \((x^3)^{1/2}\).
• This means take \(x\), raise it to the third power, and then take the square root.
• Similarly, \(x^{1/3}\) represents the cube root of \(x\).

Combining exponents through multiplication or division often requires the use of the product rule, which helps manage complex expressions.

Simplify Expressions

Simplifying expressions involves reducing complex terms to their most basic form. This process makes calculations easier and more manageable. To simplify, use available rules such as the product rule, which helps with multiplying exponents.
When simplifying expressions with roots and exponents, you can often condense multiple steps into a single expression. For example:

• Rewriting \(\sqrt{x^3}\) as \(x^{3/2}\) simplifies subsequent calculations.
• Using the product rule, \(x^{1/3} * x^{1/3} = x^{2/3}\).

Through simplification, the expressions become easier to handle both algebraically and in numerical applications, which is essential for solving equations efficiently. Properly simplified expressions lead to clearer, more concise results in mathematical calculations.