If M = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}} and x satisfies the exponential equat… — State PSC Exam

If $M = \sqrt{7 + 2\sqrt{10}} + \sqrt{7 - 2\sqrt{10}}$ and $x$ satisfies the exponential equation $\left(\sqrt[3]{4}\right)^{2x-1} = \left(\frac{1}{16}\right)^{x-4}$ with $\left(\sqrt[3]{4}\right)^{2x-1} = 2^k$ , what is the value of $M^2 + k$ ?

Answer: 23.5

Answer

The value of

M^2 + k

is 23.5.

By writing

7 \pm 2\sqrt{10}

(\sqrt{5} \pm \sqrt{2})^2

, the radical simplifies cleanly to

M = 2\sqrt{5}

, giving

M^2 = 20

. Rewriting the index equation in terms of base 2 yields

\frac{4x-2}{3} = -4x + 16

, which gives

x = \frac{25}{8}

and exponent

k = 3.5

. Adding

M^2

and

k

results in 23.5.

Step-by-Step Solution

Simplify the nested surd expression for M

M = 2\sqrt{5}, so M^2 = 20

Recognize that

7 \pm 2\sqrt{10} = (\sqrt{5} \pm \sqrt{2})^2

Convert both sides of the exponential equation to base 2

2^{\frac{4x-2}{3}} = 2^{-4x+16}

Apply laws of indices:

\sqrt[3]{4} = 2^{2/3}

and

\frac{1}{16} = 2^{-4}

Solve for x by equating the powers of 2

x = \frac{25}{8} = 3.125

Since bases are equal, the powers must be equal.

Determine the exponent value k

k = 3.5

Substitute x into the exponent expression

k = -4x + 16

Calculate the final combined expression M^2 + k

23.5

Add

M^2 = 20

and

k = 3.5

Key Concept

Nested radical simplification using binomial square expansion combined with solving exponential equations via prime base unification.

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