`(root(3)(a^(8)))/(sqrt(a^(3)))=a^(x)` where `agt1`
In the equation above, what is the value of x?

Answer

To solve the equation \(\frac{\sqrt[3]{a^8}}{\sqrt{a^3}} = a^x\), we will simplify the left-hand side step by step and find the value of \(x\).

### Step 1: Rewrite the roots in exponent form
The cube root of \(a^8\) can be expressed as:
\[ \sqrt[3]{a^8} = a^{\frac{8}{3}} \]
The square root of \(a^3\) can be expressed as:
\[ \sqrt{a^3} = a^{\frac{3}{2}} \]

### Step 2: Substitute the exponent forms into the equation
Now we can rewrite the equation:
\[ \frac{a^{\frac{8}{3}}}{a^{\frac{3}{2}}} = a^x \]

### Step 3: Apply the property of exponents
Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify the left-hand side:
\[ a^{\frac{8}{3} - \frac{3}{2}} = a^x \]

### Step 4: Find a common denominator
To subtract \(\frac{8}{3}\) and \(\frac{3}{2}\), we need a common denominator. The least common multiple of 3 and 2 is 6. We can convert both fractions:
\[ \frac{8}{3} = \frac{16}{6} \quad \text{(multiply numerator and denominator by 2)} \]
\[ \frac{3}{2} = \frac{9}{6} \quad \text{(multiply numerator and denominator by 3)} \]

### Step 5: Subtract the fractions
Now we can perform the subtraction:
\[ \frac{16}{6} - \frac{9}{6} = \frac{16 - 9}{6} = \frac{7}{6} \]

### Step 6: Set the exponents equal
Now we have:
\[ a^{\frac{7}{6}} = a^x \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x = \frac{7}{6} \]

### Final Answer
Thus, the value of \(x\) is:
\[ \boxed{\frac{7}{6}} \]