Question

Simplify and express as positive indices: ${(xy)^{m - n}} \cdot {(yz)^{n - l}} \cdot {(zx)^{l - m}}$, then the answer is $\dfrac{{{x^l}{y^m}{z^n}}}{{{x^n}{y^l}{z^m}}} = {x^{l - n}} \cdot {y^{m - l}} \cdot {z^{n - m}}$.

Answer 1

Hint: Here indices are positive that is m, n, l > 0. To get the statement true or false we need to simplify the given problem first. First we will expand the question by separating each term and their powers, then we will put the terms containing negative powers in the denominator. After that, we will cancel similar terms (i.e. terms which have the same base and same powers) from numerator and denominator. Then we will put the terms of the denominator in the numerator (terms having the same base together) to prove the given equation.

Complete step-by-step answer:
The given problem is, ${(xy)^{m - n}} \cdot {(yz)^{n - l}} \cdot {(zx)^{l - m}}$
Where, m>0, n>0 and l>0
Expanding it into exponential fraction we get,
${x^{m - n}} \cdot {y^{m - n}} \cdot {y^{n - l}} \cdot {z^{n - l}} \cdot {z^{l - m}} \cdot {x^{l - m}}$
Separating terms of different powers and taking terms containing negative power to the denominator we get,
=$\dfrac{{{x^m}}}{{{x^n}}} \cdot \dfrac{{{y^m}}}{{{y^n}}} \cdot \dfrac{{{y^n}}}{{{y^l}}} \cdot \dfrac{{{z^n}}}{{{z^l}}} \cdot \dfrac{{{z^l}}}{{{z^m}}} \cdot \dfrac{{{x^l}}}{{{x^m}}}$
There are some equal terms in numerator and denominator.
Cancelling equal terms from numerator and denominator we get,
$\dfrac{{{x^l}{y^m}{z^n}}}{{{x^n}{y^l}{z^m}}}$
As both the numerator and denominator contain terms which has same base, putting them into one line exponential form we get,
${x^{l - n}} \cdot {y^{m - l}} \cdot {z^{n - m}}$
From the above step we got the answer is $\dfrac{{{x^l}{y^m}{z^n}}}{{{x^n}{y^l}{z^m}}} = {x^{l - n}} \cdot {y^{m - l}} \cdot {z^{n - m}}$
Hence, the statement is true.

Note: The question is an exponential equation.
Exponentiation is a mathematical operation written as ${a^n}$, involving two numbers, the base a and the exponent or power n and it is pronounced as ‘a raised to the power of n’.
You should remember all the exponential rules and formulae.
While doing exponential equations, you need to be focused about putting indices.
You need to be cautious while cancelling the same terms from numerator and denominator.