Lesson 1 · from §26, 27 of the 1910 text
Multiplying Monomials
Order doesn't matter
We assume the factors of a product can be written in any order: 2 \cdot 4 = 4 \cdot 2 and a \times b = b \times a. This is the Commutative Law of Multiplication. Just as 3 \times b is written 3b, a \times b \times c is written abc. So 2a^2 \times 3 = 6a^2, \qquad 2a \times 3b = 6ab, \qquad 4ab \cdot 3x^2 = 12abx^2.
We also assume the multiplications in a product can be grouped in any order: (3 \cdot 2)4 = 3(2 \cdot 4), and a(bc) = (ab)c. This is the Associative Law of Multiplication.
Quick check
5x \cdot 4y
Adding exponents
Since a^2 = a \cdot a and a^3 = a \cdot a \cdot a, a^2 \times a^3 = (a \cdot a)(a \cdot a \cdot a) = a^5 = a^{2+3}. Likewise b^2 \times b^3 \times b^4 = b^9, and 2ab \times 3a^2 = 6a^3b, and 4xyz \times 5xy^2 = 20x^2y^3z.
Principle. The exponent of any letter in a product equals the sum of its exponents in the factors: \;a^m \times a^n = a^{m+n}.
The sign rules from Chapter 2 still apply: +2a^2 \times (-3a^3) = -6a^5 and -2a^2 \times (-3a^3) = +6a^5.
Rule. Using the rule of signs, write the product of the coefficients, followed by every letter in either factor, each with the sum of its exponents.
A power of a monomial is just repeated multiplication: (-2y)^2 = (-2y)(-2y) = 4y^2 and (2ax^2)^3 = 8a^3x^6.
Quick check
(-3m^2)(4m^5)
From the history of algebra
Sir William Rowan Hamilton (1805–1865). Strangely, the topics in the first chapters of this book were among the last to be fully understood by mathematicians. In every science, the questions that look most obvious are often the hardest to answer: any child can ask what electricity is, but the wisest scientist can only explain what it does. Long after complicated parts of algebra were mastered, the simple laws of operating on numbers were still surrounded with haze. One of the people who did most to clear it was Hamilton, born in Dublin, Ireland. As a precocious boy of twelve he knew thirteen languages. He invented kinds of numbers that don't obey the usual laws of algebra, which threw a flood of light on the laws ordinary numbers do obey. He was the first to see the importance of the Associative Law and gave it its name.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Multiply. Try to do these in your head.
(3)(-8)
(-2)(5)
(-7)(-3)
(-4x)(3)
(-4x)^2
(7)(-5a)
(3a)(-6)
(-2y)^2
(-9a)(-10)
(-3ax)^2
(4a)(-2a)
(6abc)^2
(-11x)(2x)
(7x)(-3x)
(-2a)^3
(-2a)(-3a^2)
(5a^4)(7a^3)
(-4x)^3
(a^8)(-20a)
(-4a^5)(-6a^2)
(+6y)^3
(4x)(5y)
(-3a^2x)^2
(3x^2)(-y)
(5x^2y)(-2x^3)
(-6x^3y^2)^2
(-x^4y)(-x^2y^4)
(2ax^2)^3
(5a^3)(-4a^2)(-3a)
(3ax)(-4a^2x)(-2ax^3)