Problem 495 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\quad \sqrt[3]{250}-\sqrt[3]{54}\) (b) \(\sqrt[4]{243}-\sqrt[4]{1875}\)

Step-by-Step Solution

Answer

(a) 2\sqrt[3]{2}, (b) -2\sqrt[4]{3}

Step 1: Simplify the cube roots in part (a)

First, let's simplify each term within the cube roots. Note that \(250 = 2 \times 5^3\), and \(54 = 2 \times 3^3\).Thus, \[\sqrt[3]{250} = \sqrt[3]{2 \times 5^3} = 5\sqrt[3]{2} \and\ \sqrt[3]{54} = \sqrt[3]{2 \times 3^3} = 3\sqrt[3]{2}.\]

Step 2: Combine the simplified cube roots in part (a)

Subtract the simplified terms: \[\sqrt[3]{250} - \sqrt[3]{54} = 5\sqrt[3]{2} - 3\sqrt[3]{2} = 2\sqrt[3]{2}.\]

Step 3: Simplify the fourth roots in part (b)

For \(243 \ and \ 1875\), note \ 243 = 3^5 \ and \ 1875 = 3^1 \cdot 5^4. Therefore, \[\sqrt[4]{243} = \sqrt[4]{3^5} = 3 \cdot \sqrt[4]{3}\and\sqrt[4]{1875} = \sqrt[4]{3^1 \cdot 5^4} = 5 \cdot \sqrt[4]{3}.\]

Step 4: Combine the simplified fourth roots in part (b)

Subtract the simplified terms: \[\sqrt[4]{243} - \sqrt[4]{1875} = 3\sqrt[4]{3} - 5\sqrt[4]{3} = -2\sqrt[4]{3}.\]

Key Concepts

Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because ewline 3 × 3 × 3 = 27.ewline
To simplify a cube root, identify factors that are perfect cubes. Take \(\root[3]{250}\) as an example. Upon factorizing, 250 is composed of 2 and \(5^3\). This means we can simplify \(\root[3]{250}\) to \(5 \root[3]{2}\).ewline
If the number inside the cube root includes a perfect cube, you can extract that as a factor.

Fourth Roots

Fourth roots work similarly but involve four repetitions. The fourth root of a number is a value that, when multiplied by itself four times, gives the original number. For instance, the fourth root of 16 is 2 since 2 \(\times\) 2 \(\times\) 2 \(\times\) 2 = 16.ewline
When simplifying fourth roots, look for factors that are perfect fourth powers. Take \(\root[4]{243}\), for example. Recognize 243 is \(3^5\). We can simplify this to \(3 \root[4]{3}\).ewline
Fourth roots follow the same principle as square and cube roots: any factor that is a perfect fourth power is extracted from the radical.

Simplification of Radicals

Simplifying radicals involves breaking down the expression to its simplest form. You do this by extracting all possible perfect powers from under the radical sign. For example, in \(\root[3]{54}\), we find that 54 = \(2 \times 3^3\). This implies \(\root[3]{54} = 3 \root[3]{2}\).ewline
Here is how you can simplify radicals in steps:ewline

• Factor the number inside the radical
• Identify and separate the perfect cubes or fourth powers
• Extract these values from the radical and simplify the expression

Using this method will simplify even complex radical expressions efficiently.