Problem 23 - Intermediate Algebra

Question

Add or subtract. $$ -5 \sqrt[3]{625}+\sqrt[3]{40} $$

Step-by-Step Solution

Answer

Cannot be simplified further; expression remains as \\(-5 \cdot 5^{4/3} + \sqrt[3]{40}\\).

Step 1: Simplify each term separately

Let's start by simplifying \(-5 \sqrt[3]{625}\). We need to find the cube root of 625. Since 625 is 5 raised to the fourth power, we have \(\sqrt[3]{625} = (5^4)^{1/3} = 5^{4/3}\). Similarly, simplify \(\sqrt[3]{40}\). Since 40 cannot be expressed as a perfect cube, it remains \(\sqrt[3]{40}\).

Step 2: Combine like terms if possible

As there are no like terms between \(-5 \sqrt[3]{625}\) and \(\sqrt[3]{40}\), we cannot combine them into a single expression. Keep the expression as it is: \(-5 \cdot 5^{4/3} + \sqrt[3]{40}\).

Step 3: Evaluate each component (optional for approximation)

If needed, to evaluate numerically, calculate each term separately. \(5^{4/3}\) can be approximated using a calculator, and similarly \(\sqrt[3]{40}\) can be calculated. However, this isn't necessary for simplifying or expressing the solution in terms of radicals.

Key Concepts

Cube Roots

Understanding cube roots is essential in algebra, especially when dealing with expressions like \(-5 \sqrt[3]{625}\) and \(\sqrt[3]{40}\). A cube root of a number is a special value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2 because \(2 \times 2 \times 2 = 8\). In mathematical terms, finding the cube root of a number \(x\) is the same as raising \(x\) to the power of \(1/3\).\

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• Perfect Cubes and Simplification: Just like with square roots, cube roots are straightforward if the number is a perfect cube, like 8 or 27. For non-perfect cubes, the expression remains under the radical.
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• Example in Algebra: In the exercise, \(\sqrt[3]{625}\) simplifies to \(5^{4/3}\) because 625 is \(5^4\), showing how exponents work with roots.
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\Using these principles makes it easier to handle radicals that involve cube roots in algebraic expressions.

Simplifying Expressions

Simplifying expressions is about reducing them to their most basic form without changing their value. It's an essential skill in algebra that allows us to work more efficiently with complex equations.\

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• Why Simplify? Simplifying makes it easier to solve equations and compare expressions by expressing them in the simplest form possible.
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• Process: The process involves combining like terms, factoring, and using mathematical properties like distributive or associative laws.
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\In the exercise, we see this in the handling of terms like \(-5 \cdot \sqrt[3]{625}\) and \(\sqrt[3]{40}\). By identifying the cube root of 625 as \(5^{4/3}\), we simplify that part of the expression, even if the result doesn't combine with other terms. Understanding this step helps identify when an expression is as simple as it can be, such as realizing no further reduction is possible with \(\sqrt[3]{40}\).

Algebraic Expressions

Algebraic expressions are the foundation of algebra. They consist of numbers, variables, and operations combined to express a value or relationship. These expressions can range from simple calculations, like \(x + 5\), to more complex forms involving roots, powers, or multiple variables.\

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• Components: They typically include constants (fixed values), variables (unknowns that can change), and operators (such as addition, multiplication).
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• Evaluating Expressions: Sometimes, it's necessary to substitute numbers into an expression to find its value. Other times, expressions are combined or simplified to understand or solve equations.
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\In the given exercise, the expression \(-5 \sqrt[3]{625} + \sqrt[3]{40}\) is an example of a typical algebraic expression. It's essential to understand how each component works, particularly the radicals and coefficients, which dictate how these terms interact within the expression. By breaking down each element, the workings behind simplifying and manipulating these expressions become clearer. This foundational understanding allows students to tackle more advanced algebraic problems with confidence.