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Question : 81 of 86

Marks: +1, -0

Solve for 'b' if

b \sqrt{b} = (\sqrt{b})^{b}

1
3
1 or 3
Any positive integers

Solution:

Concept:

Rewrite the equation using exponents to solve for

b

.

Explanation:

Given:

b \sqrt{b} = (\sqrt{b})^{b}

.

Write

\sqrt{b} = b^{1/2}

.

Left side:

b \cdot b^{1/2} = b^{1 + 1/2} = b^{3/2}

.

Right side:

(b^{1/2})^{b} = b^{b/2}

.

Thus

b^{3/2} = b^{b/2}

.

For

b > 0

, equate exponents:

\frac{3}{2} = \frac{b}{2}

⇒

b = 3

.

Also check

b = 1

:

1 \cdot \sqrt{1} = 1

and

(\sqrt{1})^{1} = 1

, so

b = 1

also works.

b

must be positive for real

\sqrt{b}

, so solutions are

b = 1

or

b = 3

.

Answer:

b = 1

or

b = 3

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