Problem 36 - Elementary and Intermediate Algebra

Question

Simplify each radical expression. All variables represent positive real numbers. $$ \sqrt[3]{280 a^{5} b^{6}} $$

Step-by-Step Solution

Answer

\(2ab^2\sqrt[3]{35a^2}\)

Step 1: Factor the Radicand

First, we need to factor the expression inside the cube root, specifically 280. We factor it as follows: 280 = 2 x 2 x 2 x 5 x 7 = 2^3 x 5 x 7. Therefore, 280 = 2^3 x 5 x 7, and the expression inside the radicand becomes \(2^3 \times 5 \times 7 \times a^5 \times b^6\).

Step 2: Apply the Cube Root to the Factors

Apply the cube root to each factor separately. For instance, \(\sqrt[3]{2^3} = 2\), and we simplify \(\sqrt[3]{5}\), \(\sqrt[3]{7}\), \(\sqrt[3]{a^5}\), and \(\sqrt[3]{b^6}\).

Step 3: Simplify Variables Under the Cube Root

For \(a^5\), dividing the exponent by 3 gives us: \(\sqrt[3]{a^5} = a^{5/3} = a^{1+2/3} = a\,\sqrt[3]{a^2}\). For \(b^6\), dividing the exponent by 3 gives us: \(\sqrt[3]{b^6} = b^{6/3} = b^2\).

Step 4: Multiply the Results

Combine all simplified terms outside the radical and keep the remaining terms inside: \(2b^2a\sqrt[3]{5 \times 7 \times a^2}\). This results in \(2ab^2\sqrt[3]{35a^2}\).

Key Concepts

Cube Root

The cube root is an important mathematical concept. It helps in simplifying radical expressions. When you encounter a cube root, it refers to finding a number that, when multiplied by itself three times, gives you the original number.
For instance, finding \(\sqrt[3]{8}\) means asking: "What number multiplied by itself three times equals 8?" The answer, of course, is 2, since \(2 \times 2 \times 2 = 8\).
Working with cube roots, especially in equations, it’s crucial to separate the terms effectively to simplify them efficiently.

• Applying the cube root involves breaking down or factoring each part of the expression inside the radical.
• For perfect cubes, like \(\sqrt[3]{2^3}\), this simplifies directly to the base number (in this case, 2).

Understanding cube roots can significantly aid in solving and simplifying complex algebraic expressions.

Factoring

Factoring is a fundamental technique used to simplify expressions under radical signs. It involves breaking down numbers or algebraic expressions into their prime factors.
Take the number 280 as an example: it factors into smaller components: \(2^3 \times 5 \times 7\). This factorization helps in identifying and simplifying parts of the radical expression, especially when working with cube roots.

• Breaking down a number into prime factors makes it easier to see if the terms can be further simplified.
• This is especially useful to understand their interactions with exponents.

For algebraic expressions, factoring helps uncover hidden relationships and simplifies the radical expressions, making the solution more approachable.

Exponents

Exponents are a shorthand way to express repeated multiplication of a number by itself. Understanding how to manipulate exponents is vital when simplifying radical expressions.
For example, the expression \(a^5\) means \(a \times a \times a \times a \times a\). When dealing with cube roots, dividing the exponent by three helps simplify the expression: \(\sqrt[3]{a^5} = a^{5/3}\).

• The exponent 5 is divided by 3, resulting in a mixed number: \(a^{1+2/3}\), which translates to \(a \sqrt[3]{a^2}\).
• This breakup allows easier simplification under the radical, where excess exponent values above the cube can remain inside the radical sign.

Mastering the laws of exponents enhances your ability to simplify complex mathematical expressions.

Radicand

The radicand is the value or expression within the radical sign that we aim to simplify. In radical expressions like cube roots, correctly managing the radicand is essential.
Consider the radicand in \(\sqrt[3]{280a^5b^6}\): it consists of both numerical and variable parts. This whole setup is what you need to simplify.

• Separate the numerical factor into its smallest components, which allows you to identify perfect cubes easily.
• Manage the exponents to simplify variable parts under the cube root.

Simplifying the radicand efficiently reduces the expression into a more manageable form, easing the overall calculation process.