Sure, let's solve the given equation step-by-step using the provided values [tex]\( a = 4 \)[/tex] and [tex]\( b = \frac{9}{2} \)[/tex].
The equation provided is:
[tex]$ \frac{\sqrt{b}}{a} = a^2 \times \sqrt[4]{b} $[/tex]
1. Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a = 4, \quad b = \frac{9}{2}
\][/tex]
2. Calculate the value on the left-hand side (LHS):
[tex]\[
\frac{\sqrt{b}}{a}
\][/tex]
- Find [tex]\(\sqrt{b}\)[/tex]:
[tex]\[
b = \frac{9}{2} \implies \sqrt{b} = \sqrt{\frac{9}{2}} = \frac{\sqrt{9}}{\sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2}
\][/tex]
- Then, divide by [tex]\(a\)[/tex]:
[tex]\[
\frac{\frac{3\sqrt{2}}{2}}{4} = \frac{3\sqrt{2}}{2 \times 4} = \frac{3\sqrt{2}}{8}
\][/tex]
~ The numerical result of [tex]\( \frac{3\sqrt{2}}{8} \approx 0.5303300858899106 \)[/tex]
3. Calculate the value on the right-hand side (RHS):
[tex]\[
a^2 \times \sqrt[4]{b}
\][/tex]
- Find [tex]\(a^2\)[/tex]:
[tex]\[
a = 4 \implies a^2 = 4^2 = 16
\][/tex]
- Find [tex]\(\sqrt[4]{b}\)[/tex]:
[tex]\[
\sqrt[4]{b} = \sqrt[4]{\frac{9}{2}} = \left(\frac{9}{2}\right)^{1/4}
\][/tex]
~ Using approximation, [tex]\( \left(\frac{9}{2}\right)^{1/4} \approx 1.456253
- Multiply \(a^2\)[/tex] by [tex]\(\sqrt[4]{b}\)[/tex]:
[tex]\[
16 \times 1.456253 = 23.303605041951524
\][/tex]
Thus, after solving the equations with the given values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex], we have:
[tex]\[
\left( \frac{\sqrt{b}}{a}, a^2 \times \sqrt[4]{b} \right) = \left( 0.5303300858899106, 23.303605041951524 \right)
\][/tex]
