To solve the problem, let's start by understanding the given conditions and transforming them into an algebraic form that allows us to solve the equation.
Given:
[tex]\[
\sqrt[x]{a} = \sqrt[4]{b} = \sqrt[3]{c}
\][/tex]
and
[tex]\[
a \cdot b \cdot c = 1.
\][/tex]
We need to show that [tex]\(x + y + z = 0\)[/tex].
First, let's introduce a common variable [tex]\(k\)[/tex] to simplify the given expressions. We can write:
[tex]\[
\sqrt[x]{a} = \sqrt[4]{b} = \sqrt[3]{c} = k.
\][/tex]
Now, express [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] in terms of [tex]\(k\)[/tex]:
[tex]\[
a = k^x, \quad b = k^4, \quad c = k^3.
\][/tex]
Next, substitute these expressions into the product condition [tex]\(a \cdot b \cdot c = 1\)[/tex]:
[tex]\[
(k^x) \cdot (k^4) \cdot (k^3) = 1.
\][/tex]
Combine the exponents on the left-hand side:
[tex]\[
k^{x + 4 + 3} = k^7.
\][/tex]
So the equation becomes:
[tex]\[
k^7 = 1.
\][/tex]
The only way this can be true for any non-zero [tex]\(k\)[/tex] is if the exponent 7 equals 0 (because [tex]\(k^0 = 1\)[/tex] for any non-zero [tex]\(k\)[/tex]). Thus:
[tex]\[
x + 4 + 3 = 0.
\][/tex]
Simplify the sum of the exponents:
[tex]\[
x + 4 + 3 = 0 \implies x + 7 = 0 \implies x = -7.
\][/tex]
Now, since [tex]\(x\)[/tex], [tex]\(y\)[/tex], and [tex]\(z\)[/tex] are representing the exponents as per their respective roots:
[tex]\[
x = -7, \quad y = 4, \quad z = 3.
\][/tex]
Thus:
[tex]\[
x + y + z = -7 + 4 + 3 = 0.
\][/tex]
Hence, we have shown that:
[tex]\[
x + y + z = 0.
\][/tex]
This completes the proof.
