91Ó°ÊÓ

Solving equations often requires isolating the variable you are trying to solve for. This means you need to get that variable by itself on one side of the equation.
In the given problem, we start by recognizing that the cube root expression is already isolated:
\( \sqrt[3]{p-1}=2 \). Here, the expression inside the cube root, \( p - 1 \), is alone on one side of the equation. When dealing with equations, always check if the variable or expression involving the variable is by itself first. If it's not, use algebraic steps like addition, subtraction, multiplication, or division to isolate it.
By separating the expression from other numbers or variables, you simplify the next steps in solving the equation.

Cube roots are the inverse operation of cubing a number. Just like how the square root undoes a square, the cube root undoes a cube.
For instance, the cube root of 8 is 2 because 2 cubed (2\textsuperscript{3}) is 8. In the equation
\( \sqrt[3]{p-1}=2 \), we want to eliminate the cube root to simplify the equation. This is done by cubing both sides of the equation:
\( (\text{cube root}\{p-1})^3 = 2^3 \)
This step transforms the equation into:
\( p-1 = 8 \)
At this point, the equation becomes much simpler to solve.
Always remember, when you encounter a cube root in an equation, you can remove it by cubing both sides.

Algebraic manipulation involves using basic algebraic operations to simplify or solve equations. In solving \( p-1 = 8 \), we use the simple operation of addition.
To find the value of \( p \), we add 1 to both sides:
\( p - 1 + 1 = 8 + 1 \)
Which simplifies to:
\( p = 9 \)
The process of adding, subtracting, multiplying, or dividing parts of an equation to solve for a variable is fundamental in algebra. These manipulations help in systematically solving for the unknowns.
Keep practicing these steps with different types of equations to become more comfortable with algebraic manipulations.