Question

How do you simplify \[\dfrac{{{7^{\dfrac{1}{7}}}}}{{{7^{ - \dfrac{{13}}{7}}}}}\] leaving only positive exponents?

Answer 1

Hint: Here we need to leave the positive exponent \[{7^{\dfrac{1}{7}}}\] and convert \[{7^{ - \dfrac{{13}}{7}}}\] this to positive exponent. We solve this using laws of indices. There are several basic law indices, namely law of multiplication, law of division, law of bracket, the power rule, negative power and fractional power. We use these to solve the given problem.

Complete step by step solution:
Given,
 \[\dfrac{{{7^{\dfrac{1}{7}}}}}{{{7^{ - \dfrac{{13}}{7}}}}}\] ,
Can be rewritten as,
 \[{7^{\dfrac{1}{7}}} \times \dfrac{1}{{{7^{ - \dfrac{{13}}{7}}}}}\]
We have, Negative power \[ \Rightarrow {x^{ - n}} = \dfrac{1}{{{x^n}}}\] or \[ \Rightarrow \dfrac{1}{{{x^n}}} = {x^{ - n}}\]
Here on comparing with the given problem we have, \[x = 7\] and \[n = - \dfrac{{13}}{7}\] .
 \[ \Rightarrow \dfrac{1}{{{7^{ - \dfrac{{13}}{7}}}}} = {7^{ - \left( { - \dfrac{{13}}{7}} \right)}}\]
We know that the product of a negative number with negative number is positive number,
 \[ \Rightarrow \dfrac{1}{{{7^{ - \dfrac{{13}}{7}}}}} = {7^{\dfrac{{13}}{7}}}\]
Thus we have,
 \[ \Rightarrow \dfrac{{{7^{\dfrac{1}{7}}}}}{{{7^{ - \dfrac{{13}}{7}}}}} = {7^{\dfrac{1}{7}}} \times {7^{\dfrac{{13}}{7}}}\]
Thus we have expressed \[\dfrac{{{7^{\dfrac{1}{7}}}}}{{{7^{ - \dfrac{{13}}{7}}}}}\] as a positive exponent \[{7^{\dfrac{1}{7}}} \times {7^{\dfrac{{13}}{7}}}\] .
We can simplify this further if we want,
 \[ \Rightarrow {7^{\dfrac{1}{7} + \dfrac{{13}}{7}}}\]
 \[ \Rightarrow {7^{\dfrac{{1 + 13}}{7}}}\]
 \[ \Rightarrow {7^{\dfrac{{14}}{7}}}\]
 \[ \Rightarrow {7^2}\]
So, the correct answer is “Option C”.

Note: We have several laws of indices.
 \[ \bullet \] The first law: multiplication: if the two terms have the same base and are to be multiplied together their indices are added. That is \[ {x^m} \times {x^n} = {x^{m + n}}\]
 \[ \bullet \] The second law: division: If the two terms have the same base and are to be divided their indices are subtracted. That is \[ \dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}\]
 \[ \bullet \] The third law: brackets: If a term with a power is itself raised to a power then the powers are multiplied together. That is \[ {\left( {{x^m}} \right)^n} = {x^{m \times n}}\]
 \[ \bullet \] As we have the second law of indices which helps to explain why anything to the power of zero is equal to one. \[ {x^0} = 1\]
 \[ \bullet \] Negative power \[ {x^{ - n}} = \dfrac{1}{{{x^n}}}\]
 \[ \bullet \] The fractional power \[ {x^{\dfrac{m}{n}}} = \left( {\sqrt[n] {m}} \right)\] . We use these depending on the given problem. We also know that the product of two negative numbers is a positive number and the product of a negative (positive) number and a positive (negative) number is a negative number.