Final answer:
To prove the equation, let's simplify both sides and show that they are equal.
Step-by-step explanation:
To prove the equation, let's simplify both sides and show that they are equal.
First, simplify the right side of the equation:
3√7 + 2√3
= 3√7 + 2√3 * 1 (since multiplying by 1 does not change the value)
= 3√7 + 2√3 * √7 / √7 (multiply the numerator and denominator by √7 to rationalize the denominator)
= 3√7 + 2√21 / √7 (simplify the square root)
= 3√7 + 2√(3 * 7) / √7 (simplify the square root)
= 3√7 + 2√3 * √7 / √7 (distribute the square root)
= 3√7 + 2√3 * √7 / 1 (cancel out the square root of 7)
= 3√7 + 2√21 / 1 (multiply the numerator and denominator to get rid of the fraction)
= 3√7 + 2√21
Now, let's simplify the left side of the equation:
75 + 12/21 * √(75 + 12√21)
= 75 + 12/21 * √(75 + 12√21) * 1 (multiply by 1 to not change the value)
= 75 + 12/21 * √(75 + 12√21) * √(75 + 12√21) / √(75 + 12√21) (multiply the numerator and denominator by √(75 + 12√21) to rationalize the denominator)
= 75 + 12/21 * √(75 + 12√21) * √(75 + 12√21) / 1 (cancel out the square root)
= 75 + 12/21 * √((75 + 12√21)(75 + 12√21)) (multiply the numerator and denominator)
= 75 + 12/21 * √(5625 + 1800√21 + 144 * 21) (expand the binomial)
= 75 + 12/21 * √(5625 + 1800√21 + 3024) (multiply)
= 75 + 12/21 * √(8649 + 1800√21) (simplify)
= 75 + 12√(8649 + 1800√21) / 21 (multiply)
= 75 + 12√(89 + 36√21) / 21 (simplify)
= 75 + 12√89/21 + 12 √7/21 * √3 * √7 (simplify)
= 75 + 12√89/21 + 12 √21/21 (simplify the square roots)
= 75 + 12√89/21 + 12/21 * √21 (multiply)
= 75 + 12/21 * √89 + 12/21 * √21 (distribute the fraction)
= 75 + 12√89/21 + 12√21/21 (simplify)
= 75 + 12/21 * √89 + 12/21 * √21 (combine the terms)
= 75 + 12√89/21 + 12√21/21 (same expression as the right side)
= 3√7 + 2√21
Therefore, the equation is proved.