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# Prove that: $\dfrac{{{x^{m + n}} \times {x^{n + 1}} \times {x^{1 + m}}}}{{{{\left( {{x^m} \times {x^n} \times {x^1}} \right)}^2}}} = 1$

Last updated date: 13th Jun 2024
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Hint: For solving this type of question the properties of powers are always needed. So here we will use the property of multiplication and is given by ${x^a} \cdot {x^b} = {x^{a + b}}$ . So by applying this property then we will be able to solve such problems easily.

Formula used:
If the two powers have the bases same then there will be the multiplication of the powers, so we will add the exponents.
${x^a} \cdot {x^b} = {x^{a + b}}$
Here, $x$ will be the base
And $a,b$ will be the exponents.

So we have the power given as $\dfrac{{{x^{m + n}} \times {x^{n + 1}} \times {x^{1 + m}}}}{{{{\left( {{x^m} \times {x^n} \times {x^1}} \right)}^2}}} = 1$ .
Now on taking the numerator part of the LHS side of the equation, we get
$\Rightarrow {x^{m + n}} \times {x^{n + 1}} \times {x^{1 + m}}$
Here, we can see that the base of the power is the same and exponents are different. So now by using the multiplication property of the powers, we get the equation as
$\Rightarrow {x^{m + n}} \times {x^{n + 1}} \times {x^{1 + m}} = {x^{m + n + n + 1 + 1 + m}}$
And on solving the right side of the equation we will get the equation as
$\Rightarrow {x^{m + n}} \times {x^{n + 1}} \times {x^{1 + m}} = {x^{2\left( {1 + m + n} \right)}}$ , and we will name it equation $1$
Now we will solve the denominator of the LHS side of the equation, so on solving it we will get
$\Rightarrow {\left( {{x^m} \times {x^n} \times {x^1}} \right)^2} = {\left( {{x^{m + n + 1}}} \right)^2}$
And on solving furthermore, we will get the above equation as
$\Rightarrow {x^{2\left( {m + n + 1} \right)}}$ , and we will name its equation $2$ .
So on dividing the equation $1$ with the equation $2$ , we get the equation as
$\Rightarrow \dfrac{{{x^{2\left( {m + n + 1} \right)}}}}{{{x^{2\left( {m + n + 1} \right)}}}}$
And on solving the above equation, we can see there is a like term in both the numerator and denominator so on solving it we get
$\Rightarrow 1$
Therefore, from this, we can see that LHS is equal to the RHS.
Hence, the question $\dfrac{{{x^{m + n}} \times {x^{n + 1}} \times {x^{1 + m}}}}{{{{\left( {{x^m} \times {x^n} \times {x^1}} \right)}^2}}} = 1$ is proved.

Note: There are a total of mainly five important properties of power. The first one is the product of the powers. The second one is power to a power and the third one is the quotient of powers. Similarly, there are two more which are a little different and are the power of a product and the power of a quotient. So these are the five properties of the powers which are widely used in the questions.