Problem 20 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 3 \sqrt{7}-\sqrt[3]{x}+4 \sqrt{7}-3 \sqrt[3]{x} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 7 \sqrt{7} - 4 \sqrt[3]{x} \).

Step 1: Identify Like Terms

Identify the terms in the expression that are similar. Like terms in this expression are those that have the same root and the same radicand. Here, \(3 \sqrt{7}\) and \(4 \sqrt{7}\) are like terms, and \(-\sqrt[3]{x}\) and \(-3 \sqrt[3]{x}\) are also like terms.

Step 2: Combine Like Terms for Square Roots

Add the coefficients of the like terms for the square roots of 7. The expression \(3 \sqrt{7} + 4 \sqrt{7}\) results in \( (3+4) \sqrt{7} = 7 \sqrt{7}\).

Step 3: Combine Like Terms for Cube Roots

Add the coefficients of the like terms for the cube roots of \(x\). The expression \(-\sqrt[3]{x} - 3 \sqrt[3]{x}\) results in \((-1 - 3) \sqrt[3]{x} = -4 \sqrt[3]{x}\).

Step 4: Write the Final Simplified Expression

Combine the results from Steps 2 and 3 to form the simplified expression. Therefore, the expression simplifies to \( 7 \sqrt{7} - 4 \sqrt[3]{x} \).

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, a key skill is combining like terms. Like terms are terms that have the same variable raised to the same power, or in the case of radicals, they have the same radicand and index. In our exercise, we need to find terms that can be combined by examining both the operations and the components of each term. By focusing on the terms with the same radical or similar coefficients, we can effectively simplify the expression.

• Look for terms with the same radical, like \(3 \sqrt{7}\) and \(-\sqrt[3]{x}\).
• Identify the coefficients of these terms, which are the numbers directly in front of the variables or radicals.
• Add or subtract the coefficients of these like terms appropriately to combine them.

Combining these terms will simplify your expression size and make it easier to evaluate or manipulate for further operations. This fundamental process is necessary in algebra and aids in simplifying complex problems.

Radicals in Algebra

Radicals, such as square roots and cube roots, are expressions that feature a root symbol (√). In algebra, radicals introduce a layer of complexity and can often be seen in simplified expressions. Here, these forms indicate the actual root operation, and it's key to understand them properly.

• Square roots (\(\sqrt{}\)) involve finding a number which, when squared, gives the radicand.
• Cube roots (\(\sqrt[3]{}\)) mean finding a number that, when multiplied by itself three times, results in the radicand.

In our example, \(3 \sqrt{7}\) is a term with a square root, while \(-\sqrt[3]{x}\) is a cube root. When simplifying radicals, pay attention to whether you're dealing with like terms (same radicand and index) to combine them conveniently. Recognizing these relationships in expressions is crucial for effective manipulation.

Simplifying Expressions

Simplifying expressions means reducing them to the most basic, easily understood form without changing the expression's value. This process involves not only combining like terms but also applying basic arithmetic operations to clear up the expression. Taking our exercise, simplifying results in a more concise and manageable expression.Here's how you might go about simplification:

• Identify and combine like terms, as noted previously, since these will help you reduce the expression's size.
• Perform all arithmetic operations according to mathematical rules (PEMDAS/BODMAS).
• Always rewrite your expression neatly after simplification for clarity.

Eventually, from the original expression \(3 \sqrt{7} -\sqrt[3]{x} + 4 \sqrt{7} - 3 \sqrt[3]{x}\), we simplified down to \(7 \sqrt{7} - 4 \sqrt[3]{x}\). This final step reflects understanding and applying both combining like terms and managing radicals.