Why does a power of a sum not distribute across its terms?
Quick answer
A power applies to the entire sum, so squaring means multiplying the whole sum by itself, not squaring each addend separately. For example, (2+3)^2=25, but 2^2+3^2=13. Expanding (a+b)^2 gives a^2+2ab+b^2: the middle term comes from multiplying each addend by the other.
The question, the details, the next step.
Why can't I square each number in a sum separately? I thought powers distribute.
Because squaring a sum means multiplying the whole sum by itself: (a+b)^2=(a+b)(a+b). That multiplication includes cross-products between the two addends. A quick check shows the difference: (2+3)^2=5^2=25, while 2^2+3^2=4+9=13. So the proposed shortcut does not give the same result. The parentheses matter: the exponent applies to everything inside them, and multiplying the two sums produces more than just the separate squares. [S1]
Can you show where the missing part comes from in a binomial?
Use the distributive property on each term in the first pair:
(a+b)^2 = (a+b)(a+b) = a(a+b) + b(a+b) = a^2 + ab + ba + b^2 = a^2 + 2ab + b^2.
The middle terms are ab and ba; since multiplication gives ab=ba, together they make 2ab. For instance, (x+4)^2=x^2+4x+4x+16=x^2+8x+16. If x=1, the original expression is (1+4)^2=25, and the expanded form is 1+8+16=25. Leaving out 8x would change the expression; at x=1, it would give 17 instead. The middle term comes from the two ways to pair different addends. [S1]
But I can raise the factors of a product to a power, right? How do I tell the cases apart?
Yes. A product raised to a power can be rewritten by raising each factor: (ab)^2=a^2b^2. The key is the operation inside the parentheses: multiplication in (ab)^2, addition in (a+b)^2. The sum case creates cross-products when you multiply the parentheses; the product case does not create a corresponding added middle term. For example, (2·3)^2=36 and 2^2·3^2=36, while (2+3)^2=25 and 2^2+3^2=13. A useful habit is to rewrite the square as two copies first, then use the operation already inside the parentheses. [S1]
What's a quick way to check my work before I move on?
First identify whether the expression inside parentheses is a sum or a product. For a squared sum, write it as two identical parentheses, distribute, and collect the two cross-products. Then substitute small values as a check if you are unsure. For example, with (x+4)^2, setting x=1 gives 25; the expansion 1+8+16 also gives 25, while the incomplete 1+16 gives 17. A numerical check can catch an error, though it does not by itself prove an identity for every value. For the general expansion, the multiplication-and-distribution steps show why the middle term is required; the numerical example is a quick way to spot a missing term in a particular calculation. [S1]
A practical checklist
Check whether the expression inside parentheses is a sum or a product.
Rewrite a square as the expression multiplied by itself.
For a binomial sum, distribute to get both cross-products.
Combine equal middle terms, as in ab+ba=2ab.
Substitute a small number into the original and expanded forms as a check.
A few more useful answers
What is the correct expansion of (a+b)^2?
(a+b)^2=a^2+2ab+b^2. The 2ab comes from the two cross-products when you multiply (a+b) by itself.
Does this rule apply to every power of a sum?
The shortcut (a+b)^n=a^n+b^n is not generally valid. For a square, multiply the two binomials and distribute; higher powers need their own expansion rather than that shortcut.
How can I check whether a proposed expansion is missing a term?
Substitute small values for the variables into both the original expression and your expansion. If the results differ, the proposed expansion is incorrect; matching for one choice alone does not prove the identity.
Sources
References used for this guide.
- [S1] OpenStax Algebra and Trigonometry 2e: Perfect Square Trinomials openstax.org · Checked