Let's solve the given equation step-by-step:
[tex]\[
\sqrt{7} \div 49^x = \frac{1}{7}
\][/tex]
First, we need to express the terms involving roots and powers in a common base. We know:
[tex]\[
\sqrt{7} = 7^{1/2}
\][/tex]
Also, we can express 49 as a power of 7:
[tex]\[
49 = 7^2 \quad \text{so} \quad 49^x = (7^2)^x = 7^{2x}
\][/tex]
Now, substituting these into the original equation, we get:
[tex]\[
7^{1/2} \div 7^{2x} = \frac{1}{7}
\][/tex]
Using properties of exponents, specifically [tex]\(a^m \div a^n = a^{m-n}\)[/tex], we can simplify the left-hand side:
[tex]\[
7^{1/2 - 2x} = \frac{1}{7}
\][/tex]
Next, we rewrite [tex]\(\frac{1}{7}\)[/tex] as a power of 7. Since [tex]\(\frac{1}{7} = 7^{-1}\)[/tex], we now have:
[tex]\[
7^{1/2 - 2x} = 7^{-1}
\][/tex]
Since the bases (7) are the same on both sides of the equation, we can set the exponents equal to each other:
[tex]\[
\frac{1}{2} - 2x = -1
\][/tex]
To isolate [tex]\(x\)[/tex], we solve the equation step-by-step:
1. Add 1 to both sides:
[tex]\[
\frac{1}{2} - 2x + 1 = -1 + 1
\][/tex]
[tex]\[
\frac{1}{2} + 1 - 2x = 0
\][/tex]
2. Combine the constants on the left side:
[tex]\[
\frac{1}{2} + \frac{2}{2} - 2x = 0
\][/tex]
[tex]\[
\frac{3}{2} - 2x = 0
\][/tex]
3. Add [tex]\(2x\)[/tex]:
[tex]\[
\frac{3}{2} = 2x
\][/tex]
4. Divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{3}{2} \div 2 = \frac{3}{4}
\][/tex]
Thus, we find:
[tex]\[
x = \frac{3}{4}
\][/tex]
The step-by-step solution demonstrates that the value of [tex]\(x\)[/tex] that satisfies the given equation [tex]\(\sqrt{7} \div 49^x = \frac{1}{7}\)[/tex] is:
[tex]\[
x = \frac{3}{4}
\][/tex]
