Question

Find the value of each expression if \(x=2\) and \(y=-1\) \((x y)^{2}\)

Short answer

4

Step-by-step solution

Substitute the values

Substitute the values given for \(x\) and \(y\) into the expression \((xy)^2\). Given \(x = 2\) and \(y = -1\), the expression becomes \((2 \times -1)^2\).

Simplify the product inside the parentheses

Calculate the product of \(2\) and \(-1\). This gives: \((2 \times -1) = -2\). So, the expression is now \((-2)^2\).

Evaluate the square

Square the result of the product. This gives: \((-2)^2 = 4\).

Key concepts

Substitution

Substitution in algebra means replacing the variables in an expression with the values given for those variables. Think of it like filling in the blanks!
In our exercise, we are provided with the values for the variables: \( x = 2 \) and \( y = -1 \).
To solve the expression \( (xy)^2 \), we start by substituting these values into the expression.
This means plugging in \( 2 \) wherever you see an \( x \) and \( -1 \) wherever you see a \( y \).
So our expression becomes:\( (2 \times -1)^2 \). Substitution is a crucial step in solving algebra problems because it allows us to work with actual numbers instead of variables.

Simplifying Expressions

Simplifying expressions helps make them easier to understand and solve.
After substituting the values of the variables, our expression is \( (2 \times -1)^2 \).
Now, let's simplify inside the parentheses first.
The product of \(2\) and \(-1\) is \( -2 \).
So, \( (2 \times -1) \) simplifies to \( -2 \).
Our expression is now \((-2)^2\).Simplifying expressions makes complex problems easier to manage and solve.
This step ensures that each part of the expression is simplified before moving on to more complicated operations, like evaluating powers or combining terms.

Evaluating Powers

Evaluating powers means raising a number to a certain exponent.
In our simplified expression \( (-2)^2 \), we need to raise \( -2 \) to the power of 2.
This means multiplying \( -2 \) by itself: \( -2 \times -2 = 4 \).
It is important to remember the rules of exponents:

• A positive number raised to any power remains positive,
• A negative number raised to an even power becomes positive,
  For example, \( (-3)^4 = 81 \).
• A negative number raised to an odd power remains negative,
  For example, \( (-3)^3 = -27 \).
  Understanding these rules is essential for evaluating the powers correctly.