Answer:
[tex]1.\ \ p^2q = (\frac{p^5}{p^{-3}q^{-4}})^{\frac{1}{4}}[/tex]
[tex]2.\ \ pq^{\frac{3}{2}}} = (\frac{p^2q^7}{q^{4}})^{\frac{1}{2}}[/tex]
[tex]3.\ \ pq^2 = \frac{(pq^3)^{\frac{1}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]
[tex]4.\ \ p^2q^{\frac{1}{2}} = (p^6q^{\frac{3}{2}})^{\frac{1}{3}}[/tex]
Step-by-step explanation:
Required
Match each expression to their simplified form
1.
[tex](\frac{p^5}{p^{-3}q^{-4}})^{\frac{1}{4}}[/tex]
Simplify the expression in bracket by using the following law of indices;
[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]
The expression becomes
[tex](\frac{p^{5-(-3)}}{q^{-4}})^{\frac{1}{4}}[/tex]
[tex](\frac{p^{5+3}}{q^{-4}})^{\frac{1}{4}}[/tex]
[tex](\frac{p^8}{q^{-4}})^{\frac{1}{4}}[/tex]
Split the fraction in the bracket
[tex](p^8*\frac{1}{q^{-4}})^{\frac{1}{4}}[/tex]
Simplify the fraction by using the following law of indices;
[tex]\frac{1}{a^{-m}} = a^m[/tex]
The expression becomes
[tex](p^8*q^4)^{\frac{1}{4}}[/tex]
Further simplify the expression in bracket by using the following law of indices;
[tex](ab)^m = a^m * b^m[/tex]
The expression becomes
[tex](p^{8*\frac{1}{4}}\ *\ q^4*^{\frac{1}{4}})[/tex]
[tex](p^{\frac{8}{4}}\ *\ q^{\frac{4}{4}})[/tex]
[tex]p^2q[/tex]
Hence,
[tex](\frac{p^5}{p^{-3}q^{-4}})^{\frac{1}{4}} = p^2q[/tex]
2.
[tex](\frac{p^2q^7}{q^{4}})^{\frac{1}{2}}[/tex]
Simplify the expression in bracket by using the following law of indices;
[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]
The expression becomes
[tex]({p^2q^{7-4}}})^{\frac{1}{2}}[/tex]
[tex]({p^2q^3}})^{\frac{1}{2}}[/tex]
Further simplify the expression in bracket by using the following law of indices;
[tex](ab)^m = a^m * b^m[/tex]
The expression becomes
[tex]{p^{2*\frac{1}{2}}q^{3*\frac{1}{2}}}}[/tex]
[tex]pq^{\frac{3}{2}}}[/tex]
Hence,
[tex]pq^{\frac{3}{2}}} = (\frac{p^2q^7}{q^{4}})^{\frac{1}{2}}[/tex]
3.
[tex]\frac{(pq^3)^{\frac{1}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]
Simplify the numerator as thus:
[tex]\frac{p^{\frac{1}{2}} * q^3*^{\frac{1}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]
[tex]\frac{p^{\frac{1}{2}} * q^{\frac{3}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]
Simplify the denominator as thus:
[tex]\frac{p^{\frac{1}{2}} * q^{\frac{3}{2}}}{p^{\frac{-1}{2}}q^{\frac{-1}{2}}}[/tex]
Simplify the expression in bracket by using the following law of indices;
[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]
The expression becomes
[tex]p^{\frac{1}{2} - (\frac{-1}{2} )} * q^{\frac{3}{2} - (\frac{-1}{2}) }[/tex]
[tex]p^{\frac{1}{2} +\frac{1}{2} } * q^{\frac{3}{2} + \frac{1}{2} }[/tex]
[tex]p^{\frac{1+1}{2}} * q^{\frac{3+1}{2}}[/tex]
[tex]p^{\frac{2}{2}} * q^{\frac{4}{2}}[/tex]
[tex]pq^2[/tex]
Hence,
[tex]pq^2 = \frac{(pq^3)^{\frac{1}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]
4.
[tex](p^6q^{\frac{3}{2}})^{\frac{1}{3}}[/tex]
Simplify the expression in bracket by using the following law of indices;
[tex](ab)^m = a^m * b^m[/tex]
The expression becomes
[tex]p^6*^{\frac{1}{3}}\ *\ q^{\frac{3}{2}}*^{\frac{1}{3}}[/tex]
[tex]p^{\frac{6}{3}}\ *\ q^{\frac{3*1}{2*3}}[/tex]
[tex]p^2 *\ q^{\frac{3}{6}}[/tex]
[tex]p^2 *\ q^{\frac{1}{2}[/tex]
[tex]p^2q^{\frac{1}{2}[/tex]
Hence
[tex]p^2q^{\frac{1}{2}} = (p^6q^{\frac{3}{2}})^{\frac{1}{3}}[/tex]
