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Chapter 7: Problem 27

Simplify each expression. Give exact answers. $$2 \sqrt[3]{24}+\sqrt[3]{81}$$

Short Answer

Expert verified
7\sqrt[3]{3}

Step by step solution

01

Break down the cube roots

First, break down the cube roots into simpler expressions. For \(\sqrt[3]{24}\), recognize that 24 can be written as 8 * 3, and 8 is a cube: \[\sqrt[3]{24} = \sqrt[3]{8 \cdot 3} = \sqrt[3]{8} \cdot \sqrt[3]{3} = 2 \sqrt[3]{3}\]For \(\sqrt[3]{81}\), note that 81 can be written as \(3^4\): \[\sqrt[3]{81} = \sqrt[3]{3^4} = 3\sqrt[3]{3}\]
02

Substitute the simplified expressions

Substitute the simplified expressions back into the original equation: \[2 \sqrt[3]{24} + \sqrt[3]{81} = 2(2 \sqrt[3]{3}) + 3 \sqrt[3]{3}\]
03

Combine like terms

Combine the like terms: \[2(2 \sqrt[3]{3}) + 3 \sqrt[3]{3} = 4 \sqrt[3]{3} + 3 \sqrt[3]{3} = 7 \sqrt[3]{3}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebraic expressions
Algebraic expressions consist of variables, constants, and arithmetic operations\text{(like}\text addition, subtraction, multiplication, and division). These expressions are used to represent mathematical relationships and can be simplified by combining like terms and factoring \text {where needed}\text. In the problem, we started with the algebraic expression \(2 \sqrt[3]{24} + \sqrt[3]{81}\) and simplified it step by step. We broke down compound numbers into prime factors to find their cube roots more efficiently. This step-by-step approach helps in simplifying and understanding algebraic expressions better. Knowing how to handle cube roots and like terms ensures you can work with algebraic expressions accurately and efficiently.

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