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Surds and Indices Practice Test 2

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Question : 6 of 17

Marks: +1, -0

The value of

\left(\frac{x^b}{x^c}\right)^a \cdot \left(\frac{x^c}{x^a}\right)^b \cdot \left(\frac{x^a}{x^b}\right)^c

is:

1
$x^{2(a+b+c)}$
$\frac{1}{x^{a^2+b^2+c^2}}$
$x^{a^2+b^2+c^2-ab-bc-ca}$

Solution:

Concept:

Use laws of indices to simplify each factor and then multiply powers with the same base.

Explanation:

First factor:

\left(\frac{x^b}{x^c}\right)^a = x^{a(b-c)} = x^{ab-ac}

.

Second factor:

\left(\frac{x^c}{x^a}\right)^b = x^{b(c-a)} = x^{bc-ab}

.

Third factor:

\left(\frac{x^a}{x^b}\right)^c = x^{c(a-b)} = x^{ac-bc}

.

Multiply them:

x^{ab-ac} \cdot x^{bc-ab} \cdot x^{ac-bc}

.

Add the exponents:

ab-ac+bc-ab+ac-bc = 0

.

So the expression becomes

x^0 = 1

.

Answer:

1

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