Problem 20 - 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. $$ \sqrt{y^{3}} $$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{y^{3}}\) is \(y^{3/2}\).

Step 1: Identify the expression

We're given the expression \(\sqrt{y^{3}}\). The cube function is inside the square root.

Step 2: Apply the property of exponents and square root

The property of exponents and square root allows you to present the expression in a more simplified form. For any nonnegative real number \(x\), the square root of \(x^{3}\) can be written as \(x^{3/2}\). It means that the cube of a number is taking out from under the square root sign by multiplying the exponent by \(1/2\). The purpose of this operation is to reduce the expression to its simplest possible form.

Step 3: Simplify the expression

Hence, applying this property to our expression, \(\sqrt{y^{3}}\) can be simplified as \(y^{3/2}\).

Key Concepts

Product Rule

The product rule is a fundamental property in algebra that enables us to simplify expressions involving exponents. When you multiply two expressions that have the same base, you can simplify the product by adding the exponents. In our expression, if we had \(y^3 \cdot y^2\), we could simplify this using the product rule to \(y^{3+2} = y^5\). However, our initial problem \(\sqrt{y^{3}}\) does not show a direct product of similar bases, but understanding the product rule gives us a foundation for seeing patterns in exponents, which is essential when we apply exponent properties in general.

Square Root Properties

Square root properties include several rules that allow us to manipulate square roots in expressions. For instance, \(\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\), meaning that the square root of a product is equal to the product of the square roots. Another useful property tells us that the \(\sqrt{x^2} = |x|\), which represents the absolute value of \(x\) because square roots yield nonnegative results. When we work with exponents under a square root, like in our example \(\sqrt{y^{3}}\), we utilize these properties to simplify the expression to a more manageable form.

Exponent Properties

Exponent properties, or laws of exponents, are rules that describe how to handle expressions with powers. An important rule to consider in the context of our problem \(\sqrt{y^{3}}\) is the power of a power rule, which states \( (x^a)^b = x^{a \cdot b} \). This relates to our step by step solution where \(y^{3}\) being under a square root can be represented as raising \(y\) to the power of \(3/2\). To make this more digestible, suppose \(y\) is squared \( (y^2) \); taking the square root would yield \(y\), or \(y^{(2 \cdot 1/2)} = y^{1}\). By the same logic, \(y^3\) under the square root becomes \(y^{3/2}\), applying the exponent properties effectively.

Nonnegative Real Numbers

When simplifying expressions with square roots and exponents, it's important to note whether we're dealing with nonnegative real numbers. Nonnegative real numbers are all the positive numbers including zero, which means there are no negative numbers to consider. This is crucial because the square root of a real number is defined only for nonnegative real numbers in real analysis. In the given exercise, the assumption that variables represent nonnegative real numbers simplifies our process as we don't need to worry about absolute values after taking the square root, and our solution will also be nonnegative, as shown by the simplified \(y^{3/2}\).