Question

Cube root(8+ 3 root 21)+ cube root (8-3 root 21)

Answer 1

Answer:

• $\sqrt[3]{8+3\sqrt{21} } + \sqrt[3]{8-3\sqrt{21} }$ = 1

Step-by-step explanation:

Given

• $\sqrt[3]{8+3\sqrt{21} } + \sqrt[3]{8-3\sqrt{21} }$

To find

• The value of the expression

Solution

Let's rewrite the given as:

• $\sqrt[3]{3\sqrt{21} + 8} - \sqrt[3]{3\sqrt{21} - 8}$

Let the value is m and the expressions under 3rd roots are a and b, then we have

• a = 3√21 + 8, b = 3√21 - 8
• m = ∛a - ∛b

Cube the both sides:

• m³ = (∛a - ∛b)³
• m³ = a - b - 3$\sqrt[3]{a^2b}$ + 3$\sqrt[3]{ab^2}$

Simplify in bits:

• a - b = 3√21 + 8 - 3√21 + 8 = 16
• ab = (3√21 + 8)(3√21 - 8) = 9*21 - 64 = 125
• ∛ab = ∛125 = 5

So the expression becomes:

• m³ = 16 - 3$\sqrt[3]{125a} + \sqrt[3]{125b}$
• m³ = 16 - 15∛a + 15∛b
• m³ = 16 - 15(∛a - ∛b)
• m³ = 16 - 15m

Solve for m:

• m³ + 15m - 16 = 0
• m³ - 1 + 15m - 15 = 0
• (m - 1)(m² + m + 1) + 15(m - 1) = 0
• (m - 1)(m² + m + 16) = 0
• m - 1 = 0, m² + m + 16 = 0
• m = 1 is the only real root, the quadratic equation has no real roots

The answer is 1