Problem 23 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt[3]{15 a^{2}} \cdot \sqrt[3]{9 a^{4}} $$

Step-by-Step Solution

Answer

The simplified form is \( 3a^{2} \cdot \sqrt[3]{5} \).

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt[3]{15 a^{2}} \cdot \sqrt[3]{9 a^{4}} \). This involves multiplying two cube roots and expressing the result in simplest form.

Step 2: Apply Properties of Roots

Use the property of roots \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \) to combine the two radicands. Here \( n = 3 \), so we have: \[ \sqrt[3]{15 a^{2} \cdot 9 a^{4}}. \]

Step 3: Simplify Inside the Cube Root

Multiply the terms inside the cube root: \( 15 \cdot 9 = 135 \) and \( a^{2} \cdot a^{4} = a^{6} \). This gives us: \[ \sqrt[3]{135 a^{6}}. \]

Step 4: Simplify the Cube Root

Break down \( 135 \) into its prime factors: \( 135 = 3^{3} \cdot 5 \). Then, simplify: \( \sqrt[3]{3^{3} \cdot 5 \cdot a^{6}} = 3a^{2}\cdot \sqrt[3]{5} \).

Step 5: Final Simplification

Using the property of roots, the cube root of \( 3^{3} \) is \( 3 \), and the cube root of \( a^{6} \) is \( a^{2} \). Therefore, \( 3a^{2}\cdot \sqrt[3]{5} \) is in its simplest form.

Key Concepts

Cube Roots

The concept of cube roots is all about finding a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8.
In this exercise, we dealt with cube roots while simplifying the expression \( \sqrt[3]{15a^{2}} \cdot \sqrt[3]{9a^{4}} \). By combining these into one radical, we utilized the knowledge of cube roots since we looked for factors that replicate thrice within the cube root sign to simplify further.
Understanding cube roots is crucial when breaking down larger numbers into simpler components, as we did in our solution.

Multiplying Radicals

Multiplying radicals can seem tricky, but when done step-by-step, it becomes simpler.
When multiplying radicals with the same index, like cube roots, you multiply the insides, called radicands. A vital property is \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \).
In our example, \( \sqrt[3]{15a^{2}} \cdot \sqrt[3]{9a^{4}} \) was simplified by combining the inside terms under one radical: \( \sqrt[3]{15 \cdot 9 \cdot a^{2} \cdot a^{4}} \). This meant multiplying the numbers and variables directly, making the expression easier to manage. Multiplying radicals focuses on making calculations less complex and results more straightforward.

Prime Factorization

Prime factorization is breaking down a number into the smallest prime numbers that multiply together to give the original number.
For instance, with 135 in our problem, we identified its prime factors as \( 3^{3} \cdot 5 \). By expressing numbers this way, especially under radicals, it helps in simplifying them, showcasing patterns, and focusing on prime bases that directly influence the outcome.
Recognizing and using prime factorization makes dealing with complex expressions more organized and unlocks a straightforward path to simplification.

Properties of Exponents

The properties of exponents are fundamental in algebra, particularly when dealing with radicals. These properties allow us to express numbers in compact forms and simplify complex expressions.
For example, when combining \( a^{2} \cdot a^{4} \), you add the exponents (2 and 4), giving the result \( a^{6} \). This comes from the rule \( a^{m} \cdot a^{n} = a^{m+n} \).
Understanding how exponents behave helps simplify expressions crucially when they're inside radicals, allowing for the extraction of certain variables, leading to the simplification of expressions like \( \sqrt[3]{a^{6}} = a^{2} \). Mastering exponent properties aids in keeping algebraic expressions tidy and manageable.