Question

Which of the following expressions are equivalent? Justify your reasoning.Question 5

Answer 1

Step-by-step explanation:

A) Given the below expression;

`\sqrt[4]{x^3}`

The above can be written as;

`\sqrt[4]{x^3}=(x^3)^{(1)/(4)}`

Recall the below law of exponent;

`(a^n)^m=a^(nm)`

Applying the above law of exponent to the expression, we'll have;

`(x^3)^{(1)/(4)}=x^{(3)/(4)}`

B) Given the below expression;

`(1)/(x^(-1))`

Recall the below law of exponent;

`(1)/(a)=a^(-1)`

Applying the above law of exponent to the expression, we'll have;

`(1)/(x^(-1))=x^(-1(-1))=x^1=x`

C) Given the below expression;

`\sqrt[10]{x^5\cdot x^4\cdot x^2}`

Recall the below laws of exponents;

`\begin{gathered} a^m\cdot a^n=a^(m+n) \\ (a^n)^m=a^{^(nm)} \end{gathered}`

Applying the above law of exponent to the expression, we'll have;

`\begin{gathered} \sqrt[10]{x^(5+4+2)}=\sqrt[10]{x^(11)}^{}^{} \\ =(x^(11))^{(1)/(10)}=x^{(11)/(10)}^{}^{} \end{gathered}`

D) Given the below expression;

`x^{(1)/(3)}\cdot x^{(1)/(3)}\cdot x^{(1)/(3)}`

Recall the below law of exponents;

`a^n\cdot a^m=a^(n+m)`

Applying the above law of exponent to the expression, we'll have;

`x^{(1)/(3)}\cdot x^{(1)/(3)}\cdot x^{(1)/(3)}=x^{(1)/(3)+(1)/(3)+(1)/(3)}=x^{(1+1+1)/(3)}=x^{(3)/(3)}=x^1=x`

We can see from the above that the below expressions are equivalent because they both yield the same result as x;

`\begin{gathered} A)\sqrt[4]{x^3} \\ D)x^{(1)/(3)}\cdot x^{(1)/(3)}\cdot x^{(1)/(3)} \end{gathered}`