Problem 14 - College Algebra

Question

Use the product rule to simplify the expressions in Exercises \(7-16 .\) In Exercises \(11-16,\) assume that variables represent nonnegative real numbers. $$\sqrt{y^{3}}$$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt{y^{3}} \) is \( y^{1.5} \).

Step 1: Power of a power

Firstly, let's rewrite the the terms in the expression \( \sqrt{y^{3}} \) in terms of powers. The square root of a number is the same as raising that number to the power of \(0.5\) or \(1/2\). Hence, we can rewrite \( \sqrt{y^{3}} = y^3^{1/2} \).

Step 2: Apply the power of a power rule

In the second step, we apply the power of a power law, which is \( (a^m)^n = a^{mn} \). Applying this rule to our expression, we get \( y^3^{1/2} = y^{3*1/2} \).

Step 3: Simplify the expression

Lastly, simplify the expression. \( y^{3*1/2} = y^{1.5}\). Therefore, the simplified form of \( \sqrt{y^{3}} \) is \( y^{1.5}\).

Key Concepts

Product Rule for Radicals

Understanding the product rule for radicals is essential for students dealing with algebraic expressions involving roots. Simply put, when you have two radicals multiplied together, under certain conditions, you can multiply the radicands (the numbers or expressions inside the radical) and place them under a single radical.

Here's the general form: \[\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\]The product rule is particularly handy because it helps simplify complex radical expressions into a single, more manageable radical. However, it's crucial to bear in mind that the variables involved should represent nonnegative real numbers to avoid any ambiguity with roots of negative numbers, as was presumed in the given exercise.

But how do we apply the product rule when the radicals are not explicitly in front of us, such as in the exercise with \(\sqrt{y^{3}}\)? This is where the concept of representing roots as exponents comes into play.

Understanding Exponents

Exponents can seem daunting at first, but they are simply a way to express repeated multiplication. For example, \(y^3\) means \(y \times y \times y\). An important aspect of exponents is understanding how to manipulate them when they are a part of radical expressions. Radicals, like square roots, can be expressed as exponents with fractional powers.

Why Fractional Exponents?

A square root is the same as raising a number to the power of \(0.5\) or \(1/2\). So, in the exercise, \(\sqrt{y^{3}}\) becomes \(y^{3 \cdot 1/2}\).

This expression represents the same underlying mathematical concept and allows us to apply other rules of exponents, like the 'power of a power' rule, to simplify the expression further. As a rule of thumb, when simplifying expressions with radicals and exponents, remember that rational exponents are just another way of expressing roots—this will almost always come in handy.

Power of a Power Rule

The power of a power rule is a crucial exponent rule which states that when you raise a power to another power, you can simplify by multiplying the exponents. This rule is represented by the equation \((a^m)^n = a^{mn}\).

In the context of the given exercise, we see this rule in action when \(\sqrt{y^{3}}\) is converted into \(y^{3^{1/2}}\). Applying the power of a power rule, the exponents are multiplied: \(3 \cdot \frac{1}{2}\), which equals \(1.5\) or \(\frac{3}{2}\). Hence, \(y^{3^{1/2}} = y^{1.5}\).

This process can be applied to any similar expression and is especially useful in simplifying radicals. It helps convert a seemingly complex radical into a simple exponential form that is often easier to work with, especially when combining like terms or further manipulating algebraic expressions. Remember, practicing how to apply the power of a power rule along with other exponent rules will improve your confidence in handling algebraic expressions effortlessly.