Question:sqrt[3]{x + 15} = 5For the value of x that satisfies the equation above, what is the value of x...

1. TRANSLATE the problem information

• Given equation: \(\sqrt[3]{\mathrm{x + 15}} = 5\)
• Need to find: \(\mathrm{x - 10}\) (not x itself!)

2. INFER the solution strategy

• To solve for x, we need to eliminate the cube root
• Since we have \(\sqrt[3]{\mathrm{something}} = 5\), we can cube both sides
• This uses the property that \((\sqrt[3]{\mathrm{a}})^3 = \mathrm{a}\)

3. SIMPLIFY by cubing both sides

• \((\sqrt[3]{\mathrm{x + 15}})^3 = 5^3\)
• This gives us: \(\mathrm{x + 15 = 125}\)

4. SIMPLIFY to solve for x

• Subtract 15 from both sides: \(\mathrm{x = 125 - 15}\)
• Therefore: \(\mathrm{x = 110}\)

5. TRANSLATE to answer the actual question

• The question asks for \(\mathrm{x - 10}\), not x
• Calculate: \(\mathrm{x - 10 = 110 - 10 = 100}\)

Answer: D) 100

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students solve correctly for \(\mathrm{x = 110}\), but then select the answer choice that matches x instead of \(\mathrm{x - 10}\).

They get \(\mathrm{x = 110}\) and think they're done, missing that the question asks for \(\mathrm{x - 10}\). Since 110 isn't among the answer choices, they might get confused and guess, or they might select the closest value like Choice D (100) by accident rather than by correct reasoning.

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve by taking the cube root of both sides instead of cubing both sides.

They might write \(\sqrt[3]{\sqrt[3]{\mathrm{x + 15}}} = \sqrt[3]{5}\), which doesn't simplify the problem at all. This leads to confusion about how to proceed, causing them to get stuck and abandon systematic solution, leading to guessing.

The Bottom Line:

This problem tests whether students can execute the correct inverse operation (cubing to undo cube root) and whether they carefully read what the question actually asks for. The strategic insight and attention to detail are just as important as the mathematical mechanics.