If `(5^(y+2))/5^(3)=1,` what is y?

Answer

To solve the equation \((5^{(y+2)})/5^{3} = 1\), we can follow these steps:

### Step 1: Apply the Law of Exponents
Using the law of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify the left side of the equation.

\[ \frac{5^{(y+2)}}{5^3} = 5^{(y+2) - 3} \]

### Step 2: Simplify the Exponent
Now, simplify the exponent:

\[ 5^{(y + 2 - 3)} = 5^{(y - 1)} \]

So, we rewrite the equation as:

\[ 5^{(y - 1)} = 1 \]

### Step 3: Rewrite 1 as a Power of 5
We know that \(1\) can be expressed as \(5^0\):

\[ 5^{(y - 1)} = 5^0 \]

### Step 4: Set the Exponents Equal
Since the bases are the same, we can set the exponents equal to each other:

\[ y - 1 = 0 \]

### Step 5: Solve for y
Now, solve for \(y\):

\[ y = 1 \]

### Final Answer
Thus, the value of \(y\) is:

\[ \boxed{1} \]

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