Problem 28 - A Graphical Approach to College Algebra

Question

If possible, simplify each radical expression. Assume that all variables represent positive real numbers. $$\sqrt[3]{250}$$

Step-by-Step Solution

Answer

\( 5 \sqrt[3]{2} \)

Step 1: Factor the Radicand

Begin by factoring the number inside the cube root, which is 250. Start by dividing by the smallest prime number, 2: \[ 250 \div 2 = 125 \].Next, factor 125, which can be divided by 5:\[ 125 \div 5 = 25 \].Continue factoring 25:\[ 25 \div 5 = 5 \].Thus, 250 can be written as:\[ 250 = 2 \times 5^3 \].

Step 2: Simplify the Cube Root

Now simplify \( \sqrt[3]{250} \) using the factorization obtained.\[ \sqrt[3]{250} = \sqrt[3]{2 \times 5^3} \].Since the cube root of \(5^3\) is 5, it can be taken outside the radical:\[ \sqrt[3]{2 \times 5^3} = 5 \times \sqrt[3]{2} \].

Step 3: Write the Simplified Expression

Finally, write down the simplified expression for the original problem:\[ \sqrt[3]{250} = 5 \sqrt[3]{2} \].

Key Concepts

Cube Roots

Cube roots help us find a number that, when multiplied by itself three times, gives the original number. This process is the reverse of cubing a number. For example, the cube root of 8 is 2 because 2 × 2 × 2 = 8.
To simplify a cube root, we look for perfect cubes within the radicand (the number inside the radical). This allows us to take numbers out of the radical easily. Understanding cube roots helps in solving equations and expressions where cubes are involved.

Prime Factorization

Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers. These are numbers greater than 1 that have no factors other than 1 and themselves, like 2, 3, 5, 7, and so on.
To factorize 250, we began with the smallest prime number, 2:

• 250 divided by 2 gives us 125.
• Then, 125 divided by 5 gives us 25.
• Finally, 25 divided by 5 gives us 5.

So, the prime factorization of 250 is 2 and three 5's, or 2 × 5^3. Recognizing this helps simplify expressions, especially when dealing with radicals.

Radical Expressions

Radical expressions involve roots, such as square roots and cube roots, of numbers or variables. Simplifying these can make equations easier to handle.
When simplifying radical expressions, we look for terms that can move outside the radical, which can often be done using prime factorization. In our example,

• the cube root of 250 becomes easier to handle when expressed as 2 × 5^3.
• The cube root of 5^3 simplifies to 5, leaving us with 5 times the cube root of 2.

This simplification process is crucial in algebra to handle complex expressions and equations efficiently.