To solve the given equations for [tex]\( c \)[/tex] and [tex]\( d \)[/tex], let's analyze each equation step by step.
Equation A:
[tex]\[
\sqrt{448 x^c} = 8 x^3 \sqrt{7 x}
\][/tex]
First, simplify the left and right sides to have comparable forms:
[tex]\[
\sqrt{448 x^c} = \sqrt{448} \cdot \sqrt{x^c}
\][/tex]
[tex]\[
8 x^3 \sqrt{7 x} = 8 x^3 \cdot \sqrt{7} \cdot \sqrt{x} = 8 \sqrt{7} x^3 \sqrt{x}
\][/tex]
Rewrite [tex]\(\sqrt{x^c}\)[/tex] as [tex]\(x^{c/2}\)[/tex] and [tex]\(\sqrt{x}\)[/tex] as [tex]\(x^{1/2}\)[/tex]:
[tex]\[
\sqrt{448} \cdot x^{c/2} = 8 \sqrt{7} x^3 x^{1/2}
\][/tex]
[tex]\[
\sqrt{448} \cdot x^{c/2} = 8 \sqrt{7} x^{3 + 1/2} = 8 \sqrt{7} x^{7/2}
\][/tex]
Equate the exponents and the constants:
[tex]\[
\sqrt{448} = 8 \sqrt{7}
\][/tex]
This equality holds because [tex]\(\sqrt{448} = \sqrt{64 \cdot 7} = 8 \sqrt{7}\)[/tex].
Now, equate the exponents:
[tex]\[
\frac{c}{2} = \frac{7}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
c = 7
\][/tex]
Equation B:
[tex]\[
\sqrt[3]{576 x^d} = 4 x \sqrt[3]{9 x^2}
\][/tex]
Simplify the left and right sides to have comparable forms:
[tex]\[
\sqrt[3]{576 x^d} = \sqrt[3]{576} \cdot \sqrt[3]{x^d}
\][/tex]
[tex]\[
4 x \sqrt[3]{9 x^2} = 4 x \cdot \sqrt[3]{9} \cdot \sqrt[3]{x^2}
\][/tex]
Rewrite [tex]\(\sqrt[3]{x^d}\)[/tex] as [tex]\(x^{d/3}\)[/tex] and [tex]\(\sqrt[3]{x^2}\)[/tex] as [tex]\(x^{2/3}\)[/tex]:
[tex]\[
\sqrt[3]{576} \cdot x^{d/3} = 4 \cdot \sqrt[3]{9} \cdot x \cdot x^{2/3} = 4 \sqrt[3]{9} x^{1 + 2/3} = 4 \sqrt[3]{9} x^{5/3}
\][/tex]
Equate the exponents and the constants:
[tex]\[
\sqrt[3]{576} = 4 \sqrt[3]{9}
\][/tex]
This equality holds because [tex]\(\sqrt[3]{576} = \sqrt[3]{8^2 \cdot 9} = 8 \sqrt[3]{9}\)[/tex].
Now, equate the exponents:
[tex]\[
\frac{d}{3} = \frac{5}{3}
\][/tex]
Multiply both sides by 3:
[tex]\[
d = 7
\][/tex]
Therefore, the values of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] are:
In equation [tex]\(A\)[/tex], [tex]\(c\)[/tex] is [tex]\(7\)[/tex]. In equation [tex]\(B\)[/tex], [tex]\(d\)[/tex] is [tex]\(7\)[/tex].
