Problem 38 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{a^{2}} \sqrt{a} $$

Step-by-Step Solution

Answer

Answer: The product of the given expressions is \(\sqrt{a^{3}}\).

Step 1: Recall the rule for multiplying roots

To multiply square roots with the same radicand (the number inside the square root), we can multiply the radicands and take the square root of the product: \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\).

Step 2: Apply the rule to the given expression

Our given expression is \(\sqrt{a^{2}} \cdot \sqrt{a}\). Following the rule we recall in step 1, we get: $$\sqrt{a^{2} \cdot a}$$

Step 3: Simplify the radicand

Now, we need to simplify the expression inside the square root: \(a^{2} \cdot a\). Since we are multiplying a variable with the same base, we add their exponents: $$a^{2+1} = a^{3}$$

Step 4: Write the simplified expression

After simplification, we rewrite the whole expression: $$\sqrt{a^{3}}$$ So, the product of the given expressions is: $$\boxed{\sqrt{a^{3}}}$$

Key Concepts

Understanding Radicands

A radicand is the number or expression you find under a square root symbol. In our exercise, the radicands are \(a^2\) and \(a\). To multiply square roots, it's vital to understand what you're actually multiplying—the radicands themselves.

• When multiplying \(\sqrt{a^2}\) and \(\sqrt{a}\), you're essentially dealing with their radicands: \(a^2\) and \(a\).
• Think of the radicand as the 'content' of the square root, and when you multiply square roots, you are multiplying these 'contents.'

This process makes tasks like our exercise simpler to handle since you're directly working with the radicands. Grasping this fundamental concept allows you to skillfully perform operations involving square roots.

Simplifying Expressions

Simplifying expressions is a crucial step to make math problems easier and clearer. With square roots, simplification often involves working out the expression inside the root. In our exercise, that means simplifying \(a^2 \cdot a\).

• When you have a product of the same base, like \(a^2\) and \(a\), you can simplify by adding the exponents.
• The expression \(a^2 \cdot a\) simplifies to \(a^{2+1} = a^3\).

Simplification isn't just about making expressions shorter; it's about revealing their true form, so you can deal with them more easily. A simplified expression is often a more insightful one.

Exploring Exponents

Exponents tell us how many times a number, known as the base, is multiplied by itself. When multiplying terms with exponents that share the same base, you can simplify by adding the exponents together.
Take for instance the expression \(a^2 \cdot a\):

• Here, both parts share the base \(a\).
• Adding the exponents: \(2 + 1 = 3\), so \(a^2 \cdot a = a^3\).

Recognizing and applying these rules about exponents make it easier to handle expressions that can initially seem complex. Understanding exponents paves the way for simplifying square roots and many other algebraic operations.