Understanding Exponents and Square Roots

Learning Objectives

• Evaluate expressions containing exponents.
• Write repeated factors using exponential notation.
• Find a square root of a perfect square.

Exponents provide a special way of writing repeated multiplication. Numbers written in this way have a specific form, with each part providing important information about the number. Writing numbers using exponents can save a lot of space, too. The inverse operation of multiplication of a number by itself is called finding the square root of a number. This operationA mathematical process; the four basic operations are addition, subtraction, multiplication, and division. is helpful for problems about the area of a square.

Exponential notationA notation that represents repeated multiplication using a base and an exponent. For example, `2^4` is notation that means `2 * 2 * 2 * 2`. This notation tells you that `2` is used as a factor `4` times. `2^4=16`. (Also called exponential form.) is a special way of writing repeated factors, for example `7 * 7`. Exponential notation has two parts. One part of the notation is called the baseIn a percent problem, the base represents how much should be considered `100%` (the whole); in exponents, the base is the value that is raised to a power when a number is written in exponential notation. In the example of `5^3`, `5` is the base.. The base is the number that is being multiplied by itself. The other part of the notation is the exponent The number that indicates how many times the base is used as a factor. In the example of `5^3`, `3` is the exponent and means that `5` is used three times as a factor: `5 * 5 * 5`. , or power. This is the small number written up high to the right of the base. The exponent, or power, tells how many times to use the base as a factor A number that is multiplied by another number or numbers to get a product. For example, in the equation `4 * 5 = 20`, `4` and `5` are factors. in the multiplication. In the example, `7 * 7` can be written as `7^2` (`7` is the base and `2` is the exponent). The exponent `2` means there are two factors.

`7^2 = 7 * 7 = 49`

You can read `7^2` as “seven squared.” This is because multiplying a number by itself is called “squaringMultiplying a number by itself, or raising the number to a power of `2`. `8^2` can be read as “`8` to the second power,” “`8` to a power of `2`,” or “`8` squared.” a number.” Similarly, raising a number to a power of `3` is called “cubingRaising a number to a power of `3`. `2^3` is read “`2` to the third power” or “`2` cubed,” and means use `2` as a factor three times in the multiplication. `2^3=2 * 2 * 2 = 8`. the number.” You can read `7^3` as “seven cubed.”

You can read `2^5` as “two to the fifth power” or “two to the power of five,” or “two raised to the power of five.” Read `8^4` as “eight to the fourth power,” or “eight to the power of four,” or “eight raised to the power of four.” This format can be used to read any number written in exponential notation. In fact, while `6^3` is most commonly read “six cubed,” it can also be read “six to the third power,” or “six to the power of three,” or "six raised to the power of three.”

To find the value of a number written in exponential form, rewrite the number as repeated multiplication and perform the multiplication. Two examples are shown below.

Writing repeated multiplication in exponential notation can save time and space. Consider the example `5 * 5 * 5 * 5`. We can use exponential notation to write this repeated multiplication as `5^4`. Since `5` is being multiplied, it is written as the base. Since the base is used `4` times in the multiplication, the exponent is `4`. The expression `5 * 5 * 5 * 5` can be rewritten in shorthand exponential notation as `5^4` and is read, “five to the fourth power” or “five to the power of `4`.”

To write repeated multiplication of the same number in exponential notation, first write the number being multiplied as the base. Then count how many times that number is used in the multiplication, and write that number as the exponent. Be sure to count the numbers, not the multiplication signs, to determine the exponent.

As you saw earlier, `5^2` is called “five squared.” “Five squared” means to multiply five by itself. In mathematics, we call multiplying a number by itself “squaring” the number. We call the result of squaring a whole number a square or a perfect squareA whole number that can be expressed as a whole number raised to a power of `2`. For example, `25` is a perfect square because `25 = 5 * 5=5^2`. . A perfect square is any number that can be written as a whole number raised to the powerWhen a base has an exponent, it can be said that the base is “raised to the power” of the exponent. For example, `3^5` is read as “`3` raised to the fifth power.” of `2`. For example, `9` is a perfect square because `3^2` is `9`. A perfect square number can be represented as a square shape, as shown below. We see that `1, 4, 9, 16, 25`, and `36` are examples of perfect squares.

To square a number, multiply the number by itself. `3` squared `= 3^2=3*3=9`.

Below are some more examples of perfect squares.

The inverse operationA mathematical operation that can reverse or “undo” another operation. Addition and subtraction are inverse operations. Multiplication and division are inverse operations. of squaring a number is called finding the square rootA value that can be multiplied by itself to give the original number. For example if the original number is `9`, then `3` is its square root because `3` multiplied by itself (`3^2`, pronounced "`3` squared") equals `9`. The symbol used for a square root is called a radical sign and goes on top of the number. The square root of `9` is written as: `sqrt(9)` of a number. Finding a square root is like asking, “what number multiplied by itself will give me this number?” The square root of `25` is `5`, because `5` multiplied by itself is equal to `25`. Square roots are written with the mathematical symbol, called a radical sign The symbol used for square root and other roots. It looks like `sqrt` and the number is written under it. For example, the square root of nine is written with the radical sign: `sqrt(9)`, which looks like this: `sqrt`. The “square root of `25`” is written `sqrt25` .

Summary

Exponential notation is a shorthand way of writing repeated multiplication of the same number. A number written in exponential notation has a base and an exponent, and each of these parts provides information for finding the value of the expression. The base tells what number is being repeatedly multiplied, and the exponent tells how many times the base is used in the multiplication. Exponents `2` and `3` have special names. Raising a base to a power of `2` is called “squaring” a number. Raising a base to a power of `3` is called “cubing” a number. The inverse of squaring a number is finding the square root of a number. To find the square root of a number, ask yourself, “What number can I multiply by itself to get this number?”