Simplify the radical expression. State any restrictions on the variable.

Question

(8x)/(64x^2 -8x)=____
What are the restrictions on the variable? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1)x(=/)____ (Type an integer of a fraction. Use a comma to separate answers as needed.)
2)There are no restrictions on x.

Answer 1

To simplify the expression \(\frac{8x}{64x^2 - 8x}\), we first factor the denominator.

1. We can factor out \(8x\) from the denominator: \[ 64x^2 - 8x = 8x(8x - 1) \]

2. Thus, the original expression becomes: \[ \frac{8x}{8x(8x - 1)} \]

3. We can cancel \(8x\) from the numerator and the denominator, but we must note that this cancellation is valid only when \(8x \neq 0\), which implies \(x \neq 0\).

4. After canceling, we obtain: \[ \frac{1}{8x - 1} \]

Now, regarding the restrictions on \(x\):

• We need to ensure the denominator is not equal to zero: \[ 8x - 1 \neq 0 \implies 8x \neq 1 \implies x \neq \frac{1}{8} \]

Therefore, the restrictions on \(x\) are:

1. \(x \neq 0\)
2. \(x \neq \frac{1}{8}\)

In conclusion, the simplified expression is: \[ \frac{1}{8x - 1} \]

And the restrictions on \(x\) are:

1. \(x \neq 0, \frac{1}{8}\)