Question

Match the numerical expressions to their simplified forms

Answer 1

`1.\ \ p^2q = ((p^5)/(p^(-3)q^(-4)))^{(1)/(4)}`

`2.\ \ pq^{(3)/(2)}} = ((p^2q^7)/(q^(4)))^{(1)/(2)}`

`3.\ \ pq^2 = \frac{(pq^3)^{(1)/(2)}}{(pq)^{(-1)/(2)}}`

`4.\ \ p^2q^{(1)/(2)} = (p^6q^{(3)/(2)})^{(1)/(3)}`

Explanation:

Required

Match each expression to their simplified form

1.

`((p^5)/(p^(-3)q^(-4)))^{(1)/(4)}`

Simplify the expression in bracket by using the following law of indices;

`(a^m)/(a^n) = a^(m-n)`

The expression becomes

`((p^(5-(-3)))/(q^(-4)))^{(1)/(4)}`

`((p^(5+3))/(q^(-4)))^{(1)/(4)}`

`((p^8)/(q^(-4)))^{(1)/(4)}`

Split the fraction in the bracket

`(p^8*(1)/(q^(-4)))^{(1)/(4)}`

Simplify the fraction by using the following law of indices;

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

The expression becomes

`(p^8*q^4)^{(1)/(4)}`

Further simplify the expression in bracket by using the following law of indices;

`(ab)^m = a^m * b^m`

The expression becomes

`(p^{8*(1)/(4)}\ *\ q^4*^{(1)/(4)})`

`(p^{(8)/(4)}\ *\ q^{(4)/(4)})`

`p^2q`

Hence,

`((p^5)/(p^(-3)q^(-4)))^{(1)/(4)} = p^2q`

2.

`((p^2q^7)/(q^(4)))^{(1)/(2)}`

Simplify the expression in bracket by using the following law of indices;

`(a^m)/(a^n) = a^(m-n)`

The expression becomes

`({p^2q^(7-4)}})^{(1)/(2)}`

`({p^2q^3}})^{(1)/(2)}`

Further simplify the expression in bracket by using the following law of indices;

`(ab)^m = a^m * b^m`

The expression becomes

`{p^{2*(1)/(2)}q^{3*(1)/(2)}}}`

`pq^{(3)/(2)}}`

Hence,

`pq^{(3)/(2)}} = ((p^2q^7)/(q^(4)))^{(1)/(2)}`

3.

`\frac{(pq^3)^{(1)/(2)}}{(pq)^{(-1)/(2)}}`

Simplify the numerator as thus:

`\frac{p^{(1)/(2)} * q^3*^{(1)/(2)}}{(pq)^{(-1)/(2)}}`

`\frac{p^{(1)/(2)} * q^{(3)/(2)}}{(pq)^{(-1)/(2)}}`

Simplify the denominator as thus:

`\frac{p^{(1)/(2)} * q^{(3)/(2)}}{p^{(-1)/(2)}q^{(-1)/(2)}}`

Simplify the expression in bracket by using the following law of indices;

`(a^m)/(a^n) = a^(m-n)`

The expression becomes

`p^{(1)/(2) - ((-1)/(2) )} * q^{(3)/(2) - ((-1)/(2)) }`

`p^{(1)/(2) +(1)/(2) } * q^{(3)/(2) + (1)/(2) }`

`p^{(1+1)/(2)} * q^{(3+1)/(2)}`

`p^{(2)/(2)} * q^{(4)/(2)}`

`pq^2`

Hence,

`pq^2 = \frac{(pq^3)^{(1)/(2)}}{(pq)^{(-1)/(2)}}`

4.

`(p^6q^{(3)/(2)})^{(1)/(3)}`

Simplify the expression in bracket by using the following law of indices;

`(ab)^m = a^m * b^m`

The expression becomes

`p^6*^{(1)/(3)}\ *\ q^{(3)/(2)}*^{(1)/(3)}`

`p^{(6)/(3)}\ *\ q^{(3*1)/(2*3)}`

`p^2 *\ q^{(3)/(6)}`

`p^2 *\ q^{(1)/(2)`

`p^2q^{(1)/(2)`

Hence

`p^2q^{(1)/(2)} = (p^6q^{(3)/(2)})^{(1)/(3)}`