The following calculations are all wrong, please explain what’s gone wrong and what is the correct way to calculate them square root of 2 cross times cube root of 4 space equals cube root of 2 cross times 4 end root equals cube root of 8 square root of 4 plus 16 end root space equals space square root of 4 plus square root of 16 equals 2 plus 4 equals 6 2 square root of 3 plus 7 square root of 3 space equals left parenthesis 2 plus 7 right parenthesis square root of 3 plus 3 end root equals 9 square root of 6
The Correct Answer and Explanation is:
Let us go through each calculation and identify the mistakes, then explain the correct method and results.
1. √2 × ∛4 = ∛(2 × 4) = ∛8
What went wrong:
- You cannot combine a square root and a cube root into a single cube root.
- √2 is not the same as ∛2, and the roots cannot be multiplied directly under one radical.
Correct calculation:
- √2 ≈ 1.414
- ∛4 ≈ 1.587
- So, √2 × ∛4 ≈ 1.414 × 1.587 ≈ 2.244 (approximate value)
There is no way to simplify this to ∛8, which equals 2.
2. √(4 + 16) = √4 + √16 = 2 + 4 = 6
What went wrong:
- √(4 + 16) is not equal to √4 + √16.
- Square roots do not distribute over addition.
Correct calculation:
- √(4 + 16) = √20
- √4 + √16 = 2 + 4 = 6
- But √20 ≈ 4.472, not 6
So, √(a + b) ≠ √a + √b
3. 2√3 + 7√3 = (2 + 7)√3 + ∛3 = 9√6
What went wrong:
- 2√3 + 7√3 correctly gives 9√3
- But then you incorrectly added ∛3 and somehow got 9√6
- You cannot combine square roots and cube roots this way
Correct calculation:
- 2√3 + 7√3 = 9√3
- If ∛3 is separate, it stays separate unless there’s a reason to combine
- There’s no valid algebraic operation that turns 9√3 + ∛3 into 9√6
Summary
The major mistakes here involve:
- Misapplying rules of radicals (roots cannot be combined across types)
- Incorrect distribution of roots over addition
- Trying to simplify expressions where no simplification is possible
Correct approach:
- Keep square roots and cube roots separate unless all terms are the same type
- Only combine like radicals (same root and same radicand)
- Use properties of exponents and radicals correctly:
√a × √b = √(a × b) only if both are square roots
But √a × ∛b cannot be simplified into one root
Accurate simplification requires following these algebraic rules carefully.
