Question

directions: use this information to answer parts a and b. solve the equation \\((3x - 5)^{\frac{1}{2}} = 10\\). part a show each step of the solution. enter the numbers into the boxes to show the solution steps. (3x - 5)^(1/2) = 10 3x - 5?=? \\(\square\\) 3x?=? \\(\square\\) x?=? \\(\square\\)

Explanation:

Step1: Square both sides

To eliminate the square root (since \((3x - 5)^{\frac{1}{2}}\) is the square root of \(3x - 5\)), we square both sides of the equation \((3x - 5)^{\frac{1}{2}} = 10\). Squaring a square root cancels them out, so we get \(3x - 5 = 10^2\). Calculating \(10^2\) gives \(100\), so \(3x - 5 = 100\).

Step2: Add 5 to both sides

To isolate the term with \(x\), we add 5 to both sides of the equation \(3x - 5 = 100\). This gives \(3x - 5 + 5 = 100 + 5\), which simplifies to \(3x = 105\).

Step3: Divide by 3

To solve for \(x\), we divide both sides of the equation \(3x = 105\) by 3. This gives \(\frac{3x}{3} = \frac{105}{3}\), so \(x = 35\).

Answer:

For the first box (after \(3x - 5\)): \(= 100\) (from squaring both sides, \(10^2 = 100\))
For the second box (after \(3x\)): \(= 105\) (from adding 5 to both sides, \(100 + 5 = 105\))
For the third box (after \(x\)): \(= 35\) (from dividing by 3, \(\frac{105}{3} = 35\))

So filling in the boxes:
\(3x - 5\) \(\boldsymbol{=}\) \(\boldsymbol{100}\)
\(3x\) \(\boldsymbol{=}\) \(\boldsymbol{105}\)
\(x\) \(\boldsymbol{=}\) \(\boldsymbol{35}\)