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

From Victory to Heartbreak—What Happened at All-American Homecoming Shocked Us All!
All-American Homecoming Revealed: The Stunning Twist That Defined a Generation
Homecoming Like Never Before: This All-American Tradition Shook the Nation!

🔗 Related Articles You Might Like:

📰 Caravan of Courage: Ewok Adventure That’ll Prove Courage Trumps Danger—Don’t Miss This Heart-Pounding Quest! 📰 Ewok Caravan of Courage: An Epic Adventure That’ll Make You Fight for Hope—Click to Watch! 📰 This Caravan of Courage Captured an Ewok Journey You’ll Never Forget—Shocking and Unforgettable! 📰 Snake Pokmon Unleashed The Hidden Power Every Trainer Needs To Know 📰 Snake Skin Fashion The Hoarders Secret To Turn Heads Fasterstep Into This Glam 📰 Snake Skin Secrets Revealedis This The Futuristic Look Youve Been Searching For 📰 Snake Tattoo Meaning Revealedwhy This Slinking Symbol Deserves A Place On Your Skin 📰 Snake Tattoo Meaning Uncovering The Bold Symbolism Behind This Fire Tattoo 📰 Snake Tattoos That Burn Hearts These Bold Designs Are Unforgettable 📰 Snake Tattoos Youll Regretor Love Forever Secret Designs Inside 📰 Snakebite Piercing Exploded Discover The Bold Trend Takeover You Cant Ignore 📰 Snakebite Piercing Exposed How This No Edge Trend Is Shocking The Fashion World 📰 Snakebite Piercing The Glowing Trend Taking Real Bodies Beyond Tattoos Dont Miss Out 📰 Snakebite Survival Secret Researchers Confirm This Surprising Drink Works 📰 Snakeskin Boots That Make Heels Severe But Stay Stylish Sturdy 📰 Snakeskin Boots Thatll Make You Glow And Warn Everyonedont Miss 📰 Snaks That Go Furthershocking Reasons Youre Eating Them Every Day 📰 Snaks Youve Been Buying Wrongdiscover The Savory Secret

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