What is the numerical value of the expression \sqrt[3]{7 + 5\sqrt{2}} - \sqrt[3]{5\sqrt{2}… — GMAT

What is the numerical value of the expression $\sqrt[3]{7 + 5\sqrt{2}} - \sqrt[3]{5\sqrt{2} - 7}$ ?

Cevap: 2

The numerical value of the expression is 2.

The value of the expression is 2. This can be demonstrated either by setting the expression equal to

x

, cubing both sides to construct the cubic equation

x^3 + 3x - 14 = 0

, and factoring out the real root

x = 2

, or by recognizing that

(1 + \sqrt{2})^3 = 7 + 5\sqrt{2}

and

(\sqrt{2} - 1)^3 = 5\sqrt{2} - 7

, which simplifies the expression directly to

(1 + \sqrt{2}) - (\sqrt{2} - 1) = 2

Define variables for the two cubic terms and write the target expression as a difference.

Let

u = \sqrt[3]{7 + 5\sqrt{2}}

and

v = \sqrt[3]{5\sqrt{2} - 7}

, so the target value is

x = u - v

Grouping nested radical terms simplifies the algebraic manipulation.

Cube both sides of

x = u - v

using the algebraic identity

(u - v)^3 = u^3 - v^3 - 3uv(u - v)

x^3 = u^3 - v^3 - 3uv \cdot x

Cubing eliminates the outer radical signs on the cubed terms.

Evaluate

u^3 - v^3

and the product

uv

u^3 - v^3 = (7 + 5\sqrt{2}) - (5\sqrt{2} - 7) = 14

, and

uv = \sqrt[3]{(5\sqrt{2}+7)(5\sqrt{2}-7)} = \sqrt[3]{50 - 49} = 1

Using the difference of squares under the cube root simplifies the product term to 1.

Substitute the evaluated terms into the cubic equation and solve for the real root

x

x^3 = 14 - 3x \implies x^3 + 3x - 14 = 0 \implies (x - 2)(x^2 + 2x + 7) = 0 \implies x = 2

The quadratic factor

x^2 + 2x + 7

has negative discriminant (

4 - 28 = -24

), leaving

x = 2

as the unique real solution.

Simplifying nested radicals using cubic algebraic identities and binomial expansions

Soru