Is there a formula for root(x)(a) xx root(y)(a)? For example, sqrt(81) xx root4(81)

Jun 6, 2017

$\sqrt[x]{{a}^{m}} \times \sqrt[y]{{a}^{m}} = {a}^{\frac{m \left(x + y\right)}{x y}} = \sqrt[x y]{{a}^{m \left(x + y\right)}}$

Explanation:

There is no formula i.e. often used by people to solve such problems. However, mathematics is full of surprises and it does not mean that we cannot have a formula.

Here it is observed that in the example you have square root and fourth roots of $81$, which is itself a power of $3$ i.e. ${3}^{4}$. Hence we will attempt a formula cosidering $a = {b}^{m}$ and we attempt

$\sqrt[x]{{a}^{m}} \times \sqrt[y]{{a}^{m}}$

= ${\left({a}^{m}\right)}^{\frac{1}{x}} \times {\left({a}^{m}\right)}^{\frac{1}{y}}$

= ${a}^{\frac{m}{x}} \times {a}^{\frac{m}{y}}$

= ${a}^{\frac{m}{x} + \frac{m}{y}}$

= ${a}^{\frac{m \left(x + y\right)}{x y}}$

= $\sqrt[x y]{{a}^{m \left(x + y\right)}}$ and that is the formula.

i.e. $\sqrt[x]{{a}^{m}} \times \sqrt[y]{{a}^{m}} = {a}^{\frac{m \left(x + y\right)}{x y}} = \sqrt[x y]{{a}^{m \left(x + y\right)}}$

If $a$ is not a power than you can use $m = 1$

Using this $\sqrt{{3}^{4}} \times \sqrt{{3}^{4}}$

= ${3}^{\frac{4 \left(2 + 4\right)}{2 \times 4}}$

= ${3}^{\frac{4 \times 6}{8}}$

= ${3}^{3} = 27$

Jun 7, 2017

$27$

Explanation:

$\sqrt{81} \times \sqrt{81}$

$\therefore 9 \times \sqrt{3 \cdot 3 \cdot 3 \cdot 3}$

$\therefore \sqrt{a} \times \sqrt{a} \times \sqrt{a} \times \sqrt{a} = a$

$\therefore 9 \times 3 = 27$