For the equation `(a^(x))/(a^(y))=a^(10) and (a^(y))^(3)=a^(x)`, if `agt1`, what is the value of x?

Options

A. `5`
B. `10`
C. `15`
D. `20`

Answer

To solve the equations \(\frac{a^x}{a^y} = a^{10}\) and \((a^y)^3 = a^x\), we will follow these steps:

### Step 1: Simplify the first equation
The first equation can be simplified using the properties of exponents:
\[ \frac{a^x}{a^y} = a^{x-y} \]
Thus, we can rewrite the equation as:
\[ a^{x-y} = a^{10} \]
Since the bases are the same, we can equate the exponents:
\[ x - y = 10 \quad \text{(Equation 1)} \]

### Step 2: Simplify the second equation
The second equation can also be simplified:
\[ (a^y)^3 = a^{3y} \]
So we can rewrite the equation as:
\[ a^{3y} = a^x \]
Again, equating the exponents gives us:
\[ 3y = x \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 2 into Equation 1
Now we will substitute \(x\) from Equation 2 into Equation 1:
\[ 3y - y = 10 \]
This simplifies to:
\[ 2y = 10 \]

### Step 4: Solve for \(y\)
Now, we can solve for \(y\):
\[ y = \frac{10}{2} = 5 \]

### Step 5: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute it back into Equation 2 to find \(x\):
\[ x = 3y = 3 \times 5 = 15 \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{15} \]

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