Problem 67 - Algebra for College Students

Question

Change each radical to simplest radical form. \(2 \sqrt[3]{81}\)

Step-by-Step Solution

Answer

The simplest radical form is \(6\sqrt[3]{3}\).

Step 1: Prime Factorization

Begin by factoring 81 into its prime factors. 81 can be broken down as follows: \(81 = 3 \times 27 = 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 = 3^4\). Thus, 81 is equivalent to \(3^4\).

Step 2: Simplify the Radical

Substitute the prime factorization back into the cube root: \(\sqrt[3]{81} = \sqrt[3]{3^4}\). Since we're dealing with a cube root, we can take three 3s out of the radical: \(\sqrt[3]{3^4} = 3\sqrt[3]{3}\) because \(3^3\) becomes 3 outside the cube root, leaving a 3 inside.

Step 3: Multiply by the Coefficient

Now multiply the simplified cube root by the coefficient outside the radical. The expression becomes: \(2 \times 3\sqrt[3]{3} = 6\sqrt[3]{3}\).

Key Concepts

Cube Roots

When you think of cube roots, imagine finding a number that when multiplied by itself three times gives you the original number. For instance, in the expression \(\sqrt[3]{81}\), we're seeking a number which when cubed equals 81. Cube roots help simplify radicals by "unpacking" numbers from inside the radical symbol. To simplify them, we often use prime factorization.

In the solution example, the cube root of \(81\) is addressed using its factors. By determining that \(81\) equals \(3^4\), you then know that part of this can be taken out of the cube root. Specifically, \(\sqrt[3]{3^3}\) allows us to bring one \(3\) out, resulting in \(3\sqrt[3]{3}\). This process helps make complex radicals simpler and easier to work with.

Prime Factorization

Prime factorization is a powerful tool for dealing with radicals, including cube roots. It breaks down a number into its most basic prime number components. These are the "building blocks" of the number.

For \(81\), as shown in the exercise solution, prime factorization gives us \(81 = 3 \times 3 \times 3 \times 3\) or \(3^4\). By expressing numbers this way, we rearrange and simplify other mathematical operations. This was crucial in the solution because it helped identify which part of the number could "escape" from under the cube root symbol, in this case, one set of \(3^3\).

• Identifies the prime numbers that multiply to a given number.
• Makes simplifying radical expressions easier.
• Facilitates division of radicals in simplest form.

Radical Expressions

Radical expressions involve roots, which might include square roots, cube roots, or higher ones like fourth roots. They allow us to express something that targets specific fractional powers of numbers. Working with radicals can appear complex, but the simplification process often reveals a clearer form.

In dealing with cube roots, for instance, simplifying involves both prime factorization and understanding how many complete sets of factors fit under the radical. This means recognizing when multiples of the root value (like the "3" in cube root) can be extracted from under the radical sign.

• Simplifying radicals can involve both removing factors and multiplying results outside the radical.
• Radicals can appear in various forms, and understanding how to handle them broadly increases math proficiency.
• The ultimate goal is to express the radical in simplest, most manageable form.

In our example of \(2 \sqrt[3]{3^4}\), by simplifying, we end up with \(6\sqrt[3]{3}\), a much more manageable equation.