The special products can be used to perform selected multiplications.
On the left, we use (x+y)(x-y) = x²-y². On the right, (x-y)² = x²-2xy+y².

Use special products to evaluate each expression. 71²

Verified step by step guidance

1. Recognize that the expression 71^2 is a perfect square, which can be evaluated using the special product formula for a square of a binomial: \( (x - y)^2 = x^2 - 2xy + y^2 \).
2. Rewrite 71 as a number close to a convenient base, for example, 70 + 1, so that you can express 71^2 as \( (70 + 1)^2 \).
3. Apply the formula for the square of a sum: \( (x + y)^2 = x^2 + 2xy + y^2 \). Here, \(x = 70\) and \(y = 1\), so substitute these values into the formula.
4. Calculate each term separately: \( x^2 = 70^2 \), \( 2xy = 2 \times 70 \times 1 \), and \( y^2 = 1^2 \).
5. Combine all the terms to express 71^2 as \( 70^2 + 2 \times 70 \times 1 + 1^2 \), which simplifies the calculation using special products.

Key Concepts

Special Products Formulas

Special products are algebraic identities that simplify multiplication, such as the difference of squares (x + y)(x - y) = x² - y² and the square of a binomial (x - y)² = x² - 2xy + y². Recognizing these patterns helps quickly expand or evaluate expressions without full multiplication.

Difference of Squares

The difference of squares formula states that the product of a sum and difference of the same two terms equals the difference of their squares: (x + y)(x - y) = x² - y². This is useful for simplifying expressions and evaluating products efficiently.

Square of a Binomial

Squaring a binomial like (x - y)² expands to x² - 2xy + y². This formula helps in quickly finding the square of sums or differences without multiplying the binomial by itself step-by-step, saving time and reducing errors.