[5 marks sum]

Question 15 Marks

Show that :
$\left(\frac{x^a}{x^b}\right)^{a^2+a b+b^2} \times\left(\frac{x^b}{x^c}\right)^{b^2+b c+c^2} \times\left(\frac{x^c}{x^a}\right)^{c^2+c a+a^2}=1$.

Answer

$\left[\right.$ Hint :$\left.(a-b)\left(a^2+a b+b^2\right)=a^3-b^3\right]$

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Question 25 Marks

Show that :
$\left(\frac{x^{a^2}}{x^{b^2}}\right)^{\frac{1}{a+b}} \times\left(\frac{x^{b^2}}{x^{c^2}}\right)^{\frac{1}{b+c}} \times\left(\frac{x^{c^2}}{x^{a^2}}\right)^{\frac{1}{c+a}}=1$.

Answer

[Hint: In each of these, use : $\left.a^2-b^2=(a-b)(a+b)\right]$

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Question 35 Marks

Show that :
$\left(\frac{x^{a+b}}{x^c}\right)^{a-b} \times\left(\frac{x^{b+c}}{x^a}\right)^{b-c} \times\left(\frac{x^{c+a}}{x^b}\right)^{c-a}=1$.

Answer

[Hint: In each of these, use : $\left.a^2-b^2=(a-b)(a+b)\right]$

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Question 45 Marks

Show that :
$\left(\frac{x^a}{x^{-b}}\right)^{a-b} \times\left(\frac{x^b}{x^{-c}}\right)^{b-c} \times\left(\frac{x^c}{x^{-a}}\right)^{c-a}=1$.

Answer

[Hint: In each of these, use : $\left.a^2-b^2=(a-b)(a+b)\right]$

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Question 55 Marks

Show that :
$\left(x^{a+b}\right)^{a-b} \times\left(x^{b+c}\right)^{b-c} \times\left(x^{c+a}\right)^{c-a}=1$.

Answer

[Hint: In each of these, use : $\left.a^2-b^2=(a-b)(a+b)\right]$

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Question 65 Marks

Show that :
$\left(\frac{x^p}{x^q}\right)^r \times\left(\frac{x^q}{x^r}\right)^p \times\left(\frac{x^r}{x^p}\right)^q=1$.

Answer

$\left[\right.$ Hint : L.H.S. $\left.=\frac{x^{p r}}{x^{q r}} \times \frac{x^{q p}}{x^{p r}} \times \frac{x^{q r}}{x^{p r}}=x^{p r+q p+q r-q r-p r-p q}=x^0\right]$

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MATHS [5 marks sum]

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