Certainly! Let's solve the equation [tex]\( 5^{y+1} = \sqrt{5} \)[/tex] step-by-step to find the value of [tex]\( y \)[/tex].
### Step 1: Rewrite the equation with exponents
We start with the given equation:
[tex]\[ 5^{y+1} = \sqrt{5} \][/tex]
Recall that the square root of 5 can be rewritten using exponents:
[tex]\[ \sqrt{5} = 5^{1/2} \][/tex]
So, the equation becomes:
[tex]\[ 5^{y+1} = 5^{1/2} \][/tex]
### Step 2: Set the exponents equal to each other
Since the bases (both are 5) are the same, we can set their exponents equal to each other:
[tex]\[ y + 1 = \frac{1}{2} \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
To find [tex]\( y \)[/tex], we need to isolate it. We do this by subtracting 1 from both sides of the equation:
[tex]\[ y + 1 - 1 = \frac{1}{2} - 1 \][/tex]
Which simplifies further to:
[tex]\[ y = \frac{1}{2} - 1 \][/tex]
### Step 4: Perform the subtraction
Now, let's subtract [tex]\( 1 \)[/tex] from [tex]\( \frac{1}{2} \)[/tex]:
[tex]\[ y = \frac{1}{2} - \frac{2}{2} \][/tex]
[tex]\[ y = \frac{1 - 2}{2} \][/tex]
[tex]\[ y = \frac{-1}{2} \][/tex]
### Step 5: Conclusion
The value of [tex]\( y \)[/tex] is:
[tex]\[ y = -\frac{1}{2} \][/tex]
In decimal form, this would be:
[tex]\[ y = -0.5 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] is [tex]\(\boxed{-\frac{1}{2}}\)[/tex].
