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Let’s begin our review of arithmetic by recalling the meaning of multiplication for whole numbers (the counting numbers and zero).
In the addition
$7+7+7+7$
the number 7 is repeated as an
addend* 4
times. Therefore, we say we have
four times seven and describe it by writing
$4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7$
The raised dot between the numbers 4 and 7 indicates multiplication. The dot directs us to multiply the two numbers that it separates. In algebra, the dot is preferred over the symbol
$\times $ to denote multiplication because the letter
$x$ is often used to represent a number. Thus,
$4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7=7+7+7+7$
Quite often, a particular number will be repeated as a factor in a multiplication. For example, in the multiplication
$7\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7$
the number 7 is repeated as a factor 4 times. We describe this by writing
${7}^{4}.$ Thus,
$7\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}7={7}^{4}$
The repeated factor is the lower number (the base), and the number recording how many times the factor is repeated is the higher number (the superscript). The superscript number is called an
exponent.
For Examples 1, 2, and 3, express each product using exponents.
$3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3.$ Since 3 occurs as a factor 6 times,
$3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3={3}^{6}$
$8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8.$ Since 8 occurs as a factor 2 times,
$8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8={8}^{2}$
$5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9.$ Since 5 occurs as a factor 3 times, we have
${5}^{3}.$ Since 9 occurs as a factor 2 times, we have
${9}^{2}.$ We should see the following replacements.
$\underset{{5}^{3}}{\underbrace{5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\underset{{9}^{2}}{\underbrace{9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9}}$
Then we have
$5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9={5}^{3}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{9}^{2}$
Expand
${3}^{5}.$ The base is 3 so it is the repeated factor. The exponent is 5 and it records the number of times the base 3 is repeated. Thus, 3 is to be repeated as a factor 5 times.
${3}^{5}=3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3$
Expand
${6}^{2}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{10}^{4}.$ The notation
${6}^{2}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{10}^{4}$ records the following two facts: 6 is to be repeated as a factor 2 times and 10 is to be repeated as a factor 4 times. Thus,
${6}^{2}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{10}^{4}=6\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}6\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10$
For the following problems, express each product using exponents.
$8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8$
${8}^{3}$
$12\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}12$
$5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5$
${5}^{7}$
$1\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}1$
$3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}4$
${3}^{5}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{4}^{2}$
$8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}8\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}15\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}15\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}15\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}15$
$2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}9$
${2}^{3}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{9}^{8}$
$3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}10$
Suppose that the letters
$x$ and
$y$ are each used to represent numbers. Use exponents to express the following product.
$x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y$
${x}^{3}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{y}^{2}$
Suppose that the letters
$x$ and
$y$ are each used to represent numbers. Use exponents to express the following product.
$x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y$
For the following problems, expand each product (do not compute the actual value).
${3}^{4}$
$3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3$
${4}^{3}$
${2}^{5}$
$2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2$
${9}^{6}$
${5}^{3}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{6}^{2}$
$5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}5\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}6\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}6$
${2}^{7}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{3}^{4}$
${x}^{4}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{y}^{4}$
$x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}x\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}y$
${x}^{6}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}{y}^{2}$
For the following problems, specify all the whole number factors of each number. For example, the complete set of whole number factors of 6 is 1, 2, 3, 6.
20
$1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}4,\text{\hspace{0.17em}}5,\text{\hspace{0.17em}}10,\text{\hspace{0.17em}}20$
14
12
$1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4,\text{\hspace{0.17em}}6,\text{\hspace{0.17em}}12$
30
21
$1,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}7,\text{\hspace{0.17em}}21$
45
11
$1,\text{\hspace{0.17em}}11$
17
19
$1,\text{\hspace{0.17em}}19$
2
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