Question

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

Short answer

4

Step-by-step solution

Substitute the values of x and y

Substitute the value of \(x = 2\) and \(y = -1\) into the expression \(x^{2} y^{2}\).

Calculate \(x^{2}\)

First, calculate \(2^{2}\). Since \(2^{2} = 4\).

Calculate \(y^{2}\)

Now, calculate \((-1)^{2}\). Since \((-1)^{2} = 1\).

Multiply the results of \(x^{2}\) and \(y^{2}\)

Multiply \(4\) (the result of \(x^{2}\)) by \(1\) (the result of \(y^{2}\)). So, \( 4 \times 1 = 4 \).

Key concepts

Understanding Algebraic Substitution

Algebraic substitution involves inserting specific values into variables within an expression. In our exercise, we were given the expression \( x^2 y^2 \) with \( x = 2 \) and \( y = -1 \). To solve, we replaced \( x \) and \( y \) with these values:

• First, substitute \( x = 2 \) into the expression.
• Then, substitute \( y = -1 \) where it appears.

This step simplifies the problem as it converts the symbols into numerical values. The expression now looks like this: \( 2^2 (-1)^2 \). This process makes complex algebraic problems more manageable.

Working with Exponents

Exponents are a shorthand way of indicating repeated multiplication of a number by itself. In the provided problem, we worked with exponents:

• The expression \( x^2 \) means \( x \times x \), so substituting \( x = 2 \), we get \( 2^2 = 2 \times 2 = 4 \).
• Similarly, \( y^2 \) means \( y \times y \), so substituting \( y = -1 \), we get \( (-1)^2 = -1 \times -1 = 1 \).

Understanding how exponents work is crucial. They can affect whether a solution is positive or negative, depending on whether the exponent is even or odd. In our case, both exponents were even, so both results were positive.

Basics of Multiplication in Algebra

Multiplication of algebraic terms involves combining all the numbers once the substitutions and operations (like exponents) are handled:

• From our example, after calculating the exponents, we got two results: \( 4 \) from \( x^2 \) and \( 1 \) from \( y^2 \).
• Next, multiply these two results together: \( 4 \times 1 = 4 \).

This final step brings everything together. Remember, multiplication is associative, meaning the order in which you multiply doesn’t change the result. Understanding each individual operation helps in combining them to solve the entire expression efficiently.