Question

Simplify each expression. Assume that all variables are positive when they appear. $$5 \sqrt[3]{2}-2 \sqrt[3]{54}$$

Short answer

- \sqrt[3]{2}

Step-by-step solution

- Break Down the Radicals

First, identify the cube roots in the expression: \(5 \sqrt[3]{2} \) and \(-2 \sqrt[3]{54} \).

- Simplify the Radicals

Simplify the cube root expression by factoring inside the radicals: \( \sqrt[3]{54} = \sqrt[3]{2 \times 27} = \sqrt[3]{2} \times \sqrt[3]{27} \). We know that \( \sqrt[3]{27} = 3 \), so \( \sqrt[3]{54} \) simplifies to \( 3 \sqrt[3]{2} \).

- Substitute and Simplify

Substitute \( 3 \sqrt[3]{2} \) back into the expression: \( 5 \sqrt[3]{2} - 2(3\sqrt[3]{2}) \). This simplifies to: \( 5 \sqrt[3]{2} - 6 \sqrt[3]{2} \).

- Combine Like Terms

Combine the like terms to get the simplified answer: \( 5 \sqrt[3]{2} - 6 \sqrt[3]{2} = - \sqrt[3]{2} \).

Key concepts

cube roots

A cube root finds a value that, 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 our exercise, we are working with cube roots, which are denoted as \sqrt[3]{...}\.

Let's look at one part of the expression: \( \sqrt[3]{54} \) . To simplify it, we need to find smaller cube roots within it. By breaking down 54 into its factors, we get 54 = 2 * 27.

Since \sqrt[3]{27} = 3 \, we can replace \sqrt[3]{54} with \sqrt[3]{2} \times \sqrt[3]{27} \, and this simplifies further to 3\sqrt[3]{2}\.
Understanding cube roots is crucial for simplifying more complex radical expressions efficiently.

factoring radicals

Factoring radicals means breaking down the number under the radical sign into smaller factors to simplify it. In the given problem, we needed to simplify \( \sqrt[3]{54} \) by finding its prime factors.

For instance:

• 54 can be factored into 2 and 27 (since 54 = 2 * 27)
• We know from the cube root chart that \sqrt[3]{27} = 3 \.
• Therefore, \sqrt[3]{54} becomes \sqrt[3]{2} \times \sqrt[3]{27}\.
• This results in 3\sqrt[3]{2}\.

Factoring radicals can make solving expressions much simpler and help combine like terms later on more easily.

like terms

Like terms are terms in an expression that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients.

For example, in the expression 5\sqrt[3]{2} - 6\sqrt[3]{2}\, both terms are 'like' because they involve \( \sqrt[3]{2} \).

We combined them by simply subtracting the coefficients:\

• Coefficient of the first term is 5.


• Coefficient of the second term is -6.


• Together, they combine to give -1\sqrt[3]{2}\