Mathematics is a precise and often misunderstood language. One common mistake in algebra is the assumption that ( sqrt{a^2 b^2} a b ). This is a fallacy and can lead to incorrect results. Let's explore why this statement is incorrect and the underlying mathematical principles.

Understanding ( sqrt{a^2 b^2} )

The expression ( sqrt{a^2 b^2} ) represents the length of the hypotenuse of a right-angled triangle with legs of lengths a and b. This is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Therefore, ( a^2 b^2 ) is the value inside the square root, not ( a b ).

Algebraic Analysis of the Mistake

To further clarify why ( sqrt{a^2 b^2} ) does not simplify to ( a b ), let's analyze the given problem step-by-step:

Start with the assumption that ( sqrt{a^2 b^2} a b ). Square both sides to get ( a^2 b^2 (a b)^2 ). Expand the right side: ( a^2 b^2 a^2 2ab b^2 ). Subtract ( b^2 ) from both sides: ( a^2 a^2 2ab ). Subtract ( a^2 ) from both sides: ( 0 2ab ). Divide by 2: ( ab 0 ).

This last step reveals that for the equation to hold true, either a or b must be zero. This is a contradiction because we started with the general case where a and b can be any real numbers.

Proof by Counterexample

To further demonstrate the incorrectness, consider a specific example:

Let a 3 and b 4. Calculate ( sqrt{a^2 b^2} ): ( sqrt{3^2 4^2} sqrt{9 16} sqrt{25} 5 ). Calculate ( a b ): ( 3 4 7 ). Clearly, ( 5 eq 7 ).

The result shows that ( sqrt{3^2 4^2} eq 3 4 ), thus disproving the incorrect statement.

Rational Solutions and Generalizations

Considering rational solutions, assume ( a frac{z_1}{z_2} ) and ( b frac{z_3}{z_4} ). Repeating the algebraic simplification steps, it can be shown that the denominator must be a perfect square, leading to a Pythagorean triple. For instance:

Assume ( a frac{5}{4} ) and ( b frac{5}{3} ). Calculate: ( sqrt{left(frac{5}{4}right)^2 left(frac{5}{3}right)^2} ). Compute: ( left(frac{5}{4}right)^2 frac{25}{16} ), ( left(frac{5}{3}right)^2 frac{25}{9} ), and ( sqrt{frac{25}{16} frac{25}{9}} sqrt{frac{225 400}{144}} sqrt{frac{625}{144}} frac{25}{12} ). Thus, the expression evaluates to ( frac{25}{12} ), which is not equal to ( frac{5}{4} frac{5}{3} frac{15 20}{12} frac{35}{12} ).

These examples clearly demonstrate that the original statement is incorrect and cannot be simplified to ( a b ).

Conclusion

In conclusion, the statement ( sqrt{a^2 b^2} a b ) is mathematically incorrect, as it violates the principles of the Pythagorean theorem and basic algebraic simplifications. Understanding the mathematical underpinnings is crucial for accurate problem-solving and avoiding common misconceptions.