If x = \sqrt{10 + 2\sqrt{21}} - \sqrt{10 - 2\sqrt{21}} and 9^{x^2 - k} = \left(\frac{1}{24… — State PSC Exam

If $x = \sqrt{10 + 2\sqrt{21}} - \sqrt{10 - 2\sqrt{21}}$ and $9^{x^2 - k} = \left(\frac{1}{243}\right)^{k - 6}$ , find the value of $k$ .

Answer: 2

The value of

k

is 2.

Simplifying

x

yields

x = (\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

, which gives

x^2 = 12

. Substituting

x^2 = 12

into the exponential equation gives

9^{12 - k} = (1/243)^{k - 6}

. Expressing both sides with prime base 3 gives

3^{2(12 - k)} = 3^{-5(k - 6)}

, which simplifies to

3^{24 - 2k} = 3^{30 - 5k}

. Equating the exponents gives

24 - 2k = 30 - 5k

, which solves to

k = 2

Simplify the nested surds for

x

\sqrt{10 + 2\sqrt{21}} = \sqrt{7} + \sqrt{3}

and

\sqrt{10 - 2\sqrt{21}} = \sqrt{7} - \sqrt{3}

Nested surds of the form

\sqrt{a \pm 2\sqrt{b}}

simplify to

\sqrt{m} \pm \sqrt{n}

where

m + n = a

and

m \cdot n = b

Evaluate

x

and calculate

x^2

x = 2\sqrt{3}

and

x^2 = 12

Subtracting the two surds gives

(\sqrt{7} + \sqrt{3}) - (\sqrt{7} - \sqrt{3}) = 2\sqrt{3}

. Squaring

2\sqrt{3}

gives

(2)^2 \times 3 = 12

Express both sides of the exponential equation with base 3

3^{24 - 2k} = 3^{30 - 5k}

Since

9 = 3^2

and

243 = 3^5

9^{12 - k} = (3^2)^{12 - k} = 3^{24 - 2k}

and

(1/243)^{k - 6} = (3^{-5})^{k - 6} = 3^{30 - 5k}

Equate the powers and solve for

k

k = 2

Since bases are equal (

3 > 0, 3 \neq 1

), setting exponents equal gives

24 - 2k = 30 - 5k \implies 3k = 6 \implies k = 2

Simplification of nested square root surds

\sqrt{a \pm 2\sqrt{b}}

combined with equating exponents using prime base factorization.

Question