Sure, let's break down each expression and match it to its simplest form step-by-step.
1. Simplify [tex]\((a^6 b^{12})^{\frac{1}{3}}\)[/tex]:
- Apply the exponent [tex]\(\frac{1}{3}\)[/tex] to both [tex]\(a^6\)[/tex] and [tex]\(b^{12}\)[/tex]:
[tex]\[
(a^6 b^{12})^{\frac{1}{3}} = a^{6 \cdot \frac{1}{3}} b^{12 \cdot \frac{1}{3}} = a^2 b^4
\][/tex]
2. Simplify [tex]\(\frac{(a^5 b^3)^{\frac{1}{2}}}{(a b)^{-\frac{1}{2}}}\)[/tex]:
- First, simplify [tex]\((a^5 b^3)^{\frac{1}{2}}\)[/tex] and [tex]\((a b)^{-\frac{1}{2}}\)[/tex]:
[tex]\[
(a^5 b^3)^{\frac{1}{2}} = a^{5 \cdot \frac{1}{2}} b^{3 \cdot \frac{1}{2}} = a^{2.5} b^{1.5}
\][/tex]
[tex]\[
(a b)^{-\frac{1}{2}} = a^{-\frac{1}{2}} b^{-\frac{1}{2}}
\][/tex]
- Now, divide the two results:
[tex]\[
\frac{a^{2.5} b^{1.5}}{a^{-\frac{1}{2}} b^{-\frac{1}{2}}} = a^{2.5 - (-0.5)} b^{1.5 - (-0.5)} = a^{3} b^{2}
\][/tex]
3. Simplify [tex]\(\left(\frac{a^5}{a^{-3} b^{-4}}\right)^{\frac{1}{4}}\)[/tex]:
- First, rewrite the denominator:
[tex]\[
\frac{a^5}{a^{-3} b^{-4}} = a^5 \cdot a^3 \cdot b^4 = a^{5 + 3} b^4 = a^8 b^4
\][/tex]
- Now, apply the exponent [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
(a^8 b^4)^{\frac{1}{4}} = a^{8 \cdot \frac{1}{4}} b^{4 \cdot \frac{1}{4}} = a^2 b
\][/tex]
4. Simplify [tex]\(\left(\frac{a^3}{a b^{-6}}\right)^{\frac{1}{2}}\)[/tex]:
- First, rewrite the denominator:
[tex]\[
\frac{a^3}{a b^{-6}} = \frac{a^3}{a^1 \cdot b^{-6}} = a^{3-1} b^6 = a^2 b^6
\][/tex]
- Now, apply the exponent [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[
(a^2 b^6)^{\frac{1}{2}} = a^{2 \cdot \frac{1}{2}} b^{6 \cdot \frac{1}{2}} = a b^3
\][/tex]
Now, let’s match the expressions to their simplest forms:
[tex]\[
\left(a^6 b^{12}\right)^{\frac{1}{3}} \quad \longleftrightarrow \quad a^2 b^4
\][/tex]
[tex]\[
\frac{\left(a^5 b^3\right)^{\frac{1}{2}}}{\left( a b \right)^{-\frac{1}{2}}} \quad \longleftrightarrow \quad a^3 b^2
\][/tex]
[tex]\[
\left( \frac{a^5}{a^{-3} b^{-4}} \right)^{\frac{1}{4}} \quad \longleftrightarrow \quad a^2 b
\][/tex]
[tex]\[
\left( \frac{a^3}{a b^{-6}} \right)^{\frac{1}{2}} \quad \longleftrightarrow \quad a b^3
\][/tex]
