Question

describe and correct the error a student made in multiplying $2sqrt{7x^2}$ by $2sqrt{14x^3}$.
$2sqrt{7x^2} \cdot 2sqrt{14x^3}$
$= 2\cdot 2sqrt{7x^2 \cdot 14x^3}$
$= 4sqrt{7 \cdot 2 \cdot 7 \cdot x \cdot x \cdot x \cdot x \cdot x}$
$= 8 \cdot 7sqrt{x^2 \cdot x^2 \cdot x}$
$= 56x^2sqrt{x}$

Explanation:

Step1: Multiply coefficients and radicands

$2\sqrt{7x^2} \cdot 2\sqrt{14x^3} = (2 \cdot 2) \cdot \sqrt{7x^2 \cdot 14x^3}$

Step2: Simplify product inside radical

$= 4 \cdot \sqrt{98x^{2+3}} = 4\sqrt{98x^5}$

Step3: Factor radicand into perfect squares

$98 = 49 \cdot 2$, $x^5 = x^4 \cdot x$, so:
$= 4\sqrt{49 \cdot 2 \cdot x^4 \cdot x} = 4\sqrt{49x^4 \cdot 2x}$

Step4: Extract perfect square factors

$\sqrt{49x^4} = 7x^2$, so:
$= 4 \cdot 7x^2 \cdot \sqrt{2x} = 28x^2\sqrt{2x}$

Step5: Identify student's error

The student incorrectly factored $14x^3$ as $2 \cdot 7 \cdot x^4$ (overcounted $x$ terms) and miscalculated the coefficient product: they incorrectly multiplied $2 \cdot 2$ as 8 instead of 4, and mis-simplified the radical by overextracting $x^2$ terms, leading to an incorrect coefficient $56x^2$.

Answer:

The student made two key errors:

1. They miscalculated the product of the coefficients: $2 \cdot 2 = 4$, not 8.
2. They incorrectly factored the radicand, overcounting $x$ terms and mis-simplifying the radical.

The correct simplified product is $\boldsymbol{28x^2\sqrt{2x}}$.