Solving the Interesting Equation: Rac{(√7 + √3)²}{(√7)² − (√3)²} = (7 + 2√21 + 3)/(7 − 3) = (5 + √21)/2

Mathematics often hides elegant truths beneath layers of symbols and operations. One such intriguing relation involves radical expressions:
rac{(√7 + √3)²}{(√7)² − (√3)²} = (5 + √21)/2

In this article, we’ll break down this identity step-by-step, clarify implicit steps, and explore why this result—combining binomial expansion, algebraic simplification, and the difference of squares—is both elegant and instructive.

Understanding the Context

Step 1: Expand the Numerator — (√7 + √3)²

The expression begins with the numerator:
(√7 + √3)²

Using the algebraic identity:
(a + b)² = a² + 2ab + b²
we expand:
(√7 + √3)² = (√7)² + 2(√7)(√3) + (√3)²

rac{(\sqrt{7} + \sqrt{3})^2}{(\sqrt{7})^2 - (\sqrt{3})^2} = rac{7 + 2\sqrt{21} + 3}{7 - 3} = rac{10 + 2\sqrt{21}}{4} = rac{5 + \sqrt{21}}{2}.

Key Insights

Calculate each term:

  • (√7)² = 7
  • (√3)² = 3
  • 2(√7)(√3) = 2√(7·3) = 2√21

Thus,
(√7 + √3)² = 7 + 2√21 + 3 = 10 + 2√21

Step 2: Simplify the Denominator — (√7)² − (√3)²

The denominator is a classic difference of squares:
(√7)² − (√3)²

Continue Reading

$T_4 = S_3 = 4$
$U_4 = T_3 = 2$
$n=5$:

🔗 Related Articles You Might Like:

📰 This Basic Drawing Technique Will Transform Your Skill in Minutes 📰 The Hidden Power of a Only Sketch: How It Unlocks Better Drawing Forever 📰 You’re Missing the Secret Behind Every Masterpiece in This Base Drawing Breakdown 📰 Unlock The Secret To The Perfect Minecraft Roof That Elevates Your Build Instantly 📰 Unlock The Secret To Winning Every Time In Multiplayer Switch Games Check This Out 📰 Unlock The Secrets In This Explosive Mortal Kombat 2 Trailer You Cant Miss 📰 Unlock The Secrets Of A Chef Level Minecraft Mob Farm That Racks Up Eternity 📰 Unlock The Secrets Of A Perfect Mf Buildclick To Transform Your Game 📰 Unlock The Secrets Of Everest The Himalayas With This Stunning Detailed Map 📰 Unlock The Secrets Of Morse Code Master Alphabet Transmission Instantly 📰 Unlock The Secrets Of Mortal Kombat Characters 1 Before Its Too Late 📰 Unlock The Secrets Of Mtg Final Fantasy Before It Rules Your Next Game Night 📰 Unlock The Secrets Of Music Staff Notes In Bass Clef Master Bass Clef Like A Pro 📰 Unlock The Secrets Of Muspelheim The Hidden Fire Realms Of Norse Mythology 📰 Unlock The Secrets Of Rare Minecraft Wolf Variants Watch Them Dominate 📰 Unlock The Secrets Of The Best Mob Farm Minecraft Termites Spiders More 📰 Unlock The Secrets Of The Mortal Kombat Legacy Collection Rare Backwards Exclusive Releases Alert 📰 Unlock The Secrets To Epic Minecraft Floor Designs Double Your Build Quality Now

Final Thoughts

Apply the identity:
a² − b² = (a − b)(a + b)
but here we can directly simplify:
(√7)² = 7, (√3)² = 3
⇒ 7 − 3 = 4

So, the denominator becomes 4.

Step 3: Combine Numerator and Denominator

Now substitute the simplified forms back into the original fraction:
rac{(√7 + √3)²}{(√7)² − (√3)²} = (10 + 2√21) / 4

Factor numerator:
= [2(5 + √21)] / 4
= (5 + √21) / 2

Why This Equality Matters

This identity demonstrates a powerful fusion of:

  • Binomial expansion for radicals
  • Difference of squares formula
  • Careful algebraic simplification