Problem 40 - Algebra and Trigonometry

Question

Simplify the expression. Assume the letters denote any real numbers. \(\sqrt[3]{a^{2} b} \sqrt[3]{a^{4} b}\)

Step-by-Step Solution

Answer

The simplified expression is \( a^{2} \sqrt[3]{b^{2}} \).

Step 1: Use the Product Rule for Roots

Since we are multiplying two cube roots together, we can use the product rule for roots which states that \( \sqrt[3]{x} \sqrt[3]{y} = \sqrt[3]{xy} \). Thus, \( \sqrt[3]{a^{2} b} \sqrt[3]{a^{4} b} = \sqrt[3]{(a^{2} b)(a^{4} b)} \).

Step 2: Multiply the Inside of the Radical

Now, we multiply the expressions inside the cube root: \( (a^{2} b)(a^{4} b) = a^{2} \cdot a^{4} \cdot b \cdot b \).

Step 3: Simplify Inside the Radical Using Exponent Rules

Apply the exponent rules to simplify: \( a^{2} \cdot a^{4} = a^{(2+4)} = a^{6} \) and \( b \cdot b = b^{2} \). Thus, \( \sqrt[3]{a^{6} b^{2}} \).

Step 4: Simplify the Cube Root

The cube root of \( a^{6} \) is \( a^{2} \) because the exponent is divisible by 3, that is \( \sqrt[3]{a^{6}} = a^{6/3} = a^{2} \). The expression \( b^{2} \) cannot be simplified any further as a cube root, so it remains under the radical. Therefore, \( \sqrt[3]{a^{6} b^{2}} = a^{2} \sqrt[3]{b^{2}} \).

Step 5: Write the Final Simplified Expression

The simplified form of \( \sqrt[3]{a^{2} b} \sqrt[3]{a^{4} b} \) is therefore \( a^{2} \sqrt[3]{b^{2}} \).

Key Concepts

Product Rule for Roots

The product rule for roots allows us to combine two or more roots into a single root expression. This makes simplifying expressions involving radical expressions much easier. When you have two cube roots multiplied together, you can apply the product rule, which states that \( \sqrt[3]{x} \times \sqrt[3]{y} = \sqrt[3]{xy} \). This means you're taking the cube root of the product of the two expressions inside the radicals.
This rule helps reduce complexity. It's like combining similar items into one box to simplify the counting process. In our exercise, when you see \( \sqrt[3]{a^{2} b} \times \sqrt[3]{a^{4} b} \), applying this rule means you first multiply what's inside the radicals:

• Combine \( a^{2} \) with \( a^{4} \)
• Combine \( b \) with \( b \)

This gives \( \sqrt[3]{(a^{2}b) \cdot (a^{4}b)} \), simplifying the expression before you handle the cube root.

Exponent Rules

Exponent rules provide a framework for simplifying expressions involving powers of the same base. The basic principle here is that when multiplying expressions with the same base, you add their exponents. This is succinctly given by the rule: \( a^m \times a^n = a^{m+n} \). Understanding this rule helps you easily manage expressions with powers when facing multiplication tasks.For the problem at hand, after using the product rule for roots, you'll have: \((a^{2} \times a^{4}) \cdot (b \times b)\). Applying the exponent rule translates into:

• \( a^{2+4} = a^{6} \)
• \( b^{1+1} = b^{2} \)

These calculations help transform the problem into a simpler expression \( a^6b^2 \) that needs further reduction under the cube root.

Cube Root Simplification

Simplifying cube roots involves breaking down the expression inside the cube root into parts that can be simplified further. This often involves dividing the exponent by three since the cube root asks "what number times itself three times gives this?"In our expression, \( \sqrt[3]{a^6 b^2} \), handle each part separately:

• The cube root of \( a^6 \) involves assessing how many times 3 fits into 6. It fits exactly 2 times, simplifying to \( a^2 \).
• The term \( b^2 \) doesn’t simplify under cubic roots because 2 isn’t divisible by 3. Thus, it remains \( \sqrt[3]{b^2} \).

Therefore, the whole expression reduces to \( a^2 \sqrt[3]{b^2} \), achieving a simpler form by extracting what can be fully rooted out.