Find each product. See Examples 5 and 6.
$(a-6b{)}^2$

Verified step by step guidance

1. Recognize that the expression $(a-6b){)}^2$ is a binomial squared, which can be expanded using the formula for the square of a difference: $(x-y){)}^2=x^2-2xy+y^2$.
2. Identify the terms $x$ and $y$ in the expression: here, $x=a$ and $y=6b$.
3. Calculate the square of the first term: $a^2$.
4. Calculate twice the product of the two terms: $2\cdot a\cdot 6b=2\cdot a\cdot 6\cdot b$.
5. Calculate the square of the second term: $6^2\cdot b^2$.

Key Concepts

Binomial Squares

A binomial square is the product of a binomial multiplied by itself, expressed as (a + b)² or (a - b)². It expands to a² ± 2ab + b², where the middle term's sign depends on the binomial's sign. Recognizing this pattern simplifies squaring expressions like (a - 6b)².

Distributive Property

The distributive property allows multiplication over addition or subtraction, such as a(b + c) = ab + ac. It is essential for expanding products of binomials by multiplying each term in the first binomial by each term in the second.

Combining Like Terms

After expanding expressions, like terms (terms with the same variables and exponents) must be combined to simplify the expression. This step ensures the final answer is in its simplest form, such as combining terms involving 'ab' in the expansion.