Algebra part four - powers (indices)
multiplication
In algebra, the multiplication symbol can be left out. The division symbol can be replaced with the fraction bar. Originally, mathematicians wrote their maths equations in sentences or like an essay. They used no symbols (or very few) at all. Over the last 400 years or so, the literacy statements in mathematics have transferred to the symbolism we use in the classroom today. However, in education circles, some of the literacy content is making a return. This does not mean a return to the old ways of writing out mathematical statements, but at least an understanding of mathematics in terms of language symbols and text, one of the key competencies that teachers assess students against.
write out without multiplication signs
5 x x + 4 x y can be expressed as 5x + 4y where the multiplication symbols have been removed. Rewrite the following by removing the times sign. (the times sign is 'x' and x is the variable, in italics)
Activity 1
1) 2 x x + 4 6) 5 x q - r
2) 7 x m + 1 7) 9 x w + 9 x t
3) 1 x k - 2 8) x x y - z
4) 12 + 3 x n 9) 6 x a - 5 x b
5) p x 8 10) h x i x j
Answers: 2x+4, 7m+1, k-2, 3n+12 (notice the order-variables first), 8p, 5q-r, 9w+9t, xy-z, 6a-5b, hij.
Activity 1
1) 2 x x + 4 6) 5 x q - r
2) 7 x m + 1 7) 9 x w + 9 x t
3) 1 x k - 2 8) x x y - z
4) 12 + 3 x n 9) 6 x a - 5 x b
5) p x 8 10) h x i x j
Answers: 2x+4, 7m+1, k-2, 3n+12 (notice the order-variables first), 8p, 5q-r, 9w+9t, xy-z, 6a-5b, hij.
Powers or indices
When multiplication is repeated it's easier to use Indices (powers) to represent the expression (fig. 1. below). So instead of writing out 'p' 5 times it may be expressed as:
Many students get confused between multiplying and adding, so when multiplying the answer is not 5p. That would be obtained from adding p+p+p+p+p, not multiplying them! Referring to fig. 2 below, the number 8 is called the base and the power is called the index. The expression is called an Indice (pronounced inder-see). The expression says "8 raised to the power of 2" or "8 squared". To recap - Indices are the shorthand way of expressing numbers and variables (letters) where their multiplication has been repeat multiplied.
shorten the following expressions by writing in index form
Activity 2:
1) m x m x m x m
2) p x p
3) a x a x b x b x b
4) j x j x k x k x k x n x n
1) m x m x m x m
2) p x p
3) a x a x b x b x b
4) j x j x k x k x k x n x n
expanding and multiplying terms
Activity 3:
answers: kxkxkxk or kkkk, wxwxwxwxwxwxwxw or wwwwwwww, xxxxyxyxy
or xxyyy, txtxtxtxtxtxvxvxvxvxvxv or tttttvvvvvv.
or xxyyy, txtxtxtxtxtxvxvxvxvxvxv or tttttvvvvvv.
Activity 4:
complete the multiplication tables
complete the multiplication tables
Answers: (includes the first row of each grid, c, 2b, 5a, b etc)
Activity 5: Difficult
more substitution
Activity 6:
Answers: 1) 64, 2) 64, 3) 49, 4) 16, 5) 81, 6) 4, 7) 9, 8) 89, 9) 1/25, 10) 1/4 x 1/5 = 1/20.