Powers, Exponents and Roots

Introduction

Exponents play a large role in mathematical calculations. This chapter provides an introduction to the meaning of exponents and the calculations associated with them. Since exponents are used abundantly in all of mathematics, the basics taught in this chapter will become important building blocks for future knowledge.
The first section will explain the fundamentals of exponents, and explore squares, cubes, and higher order exponents. This section will explain how to square and cube numbers, as well as how to recognize a perfect square.
The second section will focus on exponents applied to specific types of numbers--namely, negative numbers, decimals, and fractions. Here, we will learn how to raise these base numbers to any power.
The third section will explore calculations in which the exponent is negative. It will explain the meaning of a negative exponent and how to evaluate expressions that contain negative exponents. It will also show the importance of negative exponents to the base ten system while discussing how write out any terminating decimal as a sum of single-digit numbers times powers of ten.
The fourth section will deal with roots--square roots, cube roots, and higher order roots--and fractional exponents. This section will explain what a root is and how to find a root if the answer is rational. It will also explain why we cannot take the square root (or any even root) of a negative number. This section will also explain how to raise a number to a fractional power.
The fifth section will deal with roots that are not easy to find. It first shows how to simplify a square root to make calculations easier, and it then shows how to find an approximate decimal value for a square root.
The final section will revisit the order of operations learned in the SparkNote on Operations, and revise this order to include calculations with exponents.
Overall, this chapter offers an introduction to exponents for those readers who have never been exposed to them, and an opportunity to sharpen knowledge to those who have. Exponents will play an important role in pre-algebra during area calculations. They will also be very important in future mathematical endeavors, including algebra, geometry, calculus, and higher mathematics.

Terms - Introduction

Base  -  The number that is raised to a power. In " 7⁴ ", 7 is the base.

Cube  -  A number times itself times itself. 5 cubed = 5³ = 5×5×5 = 125 .

Cube Root  -  A number that, when cubed, is equal to the given number.

Exponent  -  The power to which a number is raised; the number of times a number is multiplied. In " 7⁴ ", 7 has an exponent of 4, and 7 is multiplied 4 times (7×7×7×7) .

Fractional Exponent  -  An exponent in which the numerator is the power to which the number should be taken and the denominator is the root which should be taken.

Mean  -  The sum of two numbers, divided by 2 (sometimes called an average). This yields the number directly between the two numbers.

Negative Exponent  -  An exponent that takes the base number to the positive opposite of the exponent (the exponent with the negative sign removed), and places the result in the denominator of a fraction whose numerator is 1.

Perfect Square  -  The square of a whole number. 1, 4, 9, 16, 25, 36, 49, ... are all perfect squares.

Simplify (Square Root)  -  To remove all factors that are perfect squares from inside the square root sign and place their square roots outside the sign.

Square  -  A number times itself. 5 squared = 5² = 5×5 = 25 .

Square Root  -  A number that, when squared (multiplied by itself), is equal to the given number. 

 Squares, Cubes, and Higher Order Exponents 

 

Squares

The square of a number is that number times itself. 5 squared, denoted 5² , is equal to 5×5 , or 25. 2 squared is 2² = 2×2 = 4 . One way to remember the term "square" is that there are two dimensions in a square (height and width) and the number being squared appears twice in the calculation. In fact, the term "square" is no coincidence--the square of a number is the area of the square with sides equal to that number.
A number that is the square of a whole number is called a perfect square. 4² = 16 , so 16 is a perfect square. 25 and 4 are also perfect squares. We can list the perfect squares in order, starting with 1² : 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...

Cubes

The cube of a number is that number times itself times itself. 5 cubed, denoted 5³ , is equal to 5×5×5 , or 125. 2 cubed is 2³ = 2×2×2 = 8 . The term "cube" can be remembered because there are three dimensions in a cube (height, width, and depth) and the number being cubed appears three times in the calculation. Similar to the square, the cube of a number is the volume of the cube with sides equal to that number--this will come in handy in higher levels of math.

Exponents

The "2" in " 5² " and the "3" in " 5³ " are called exponents. An exponent indicates the number of times we must multiply the base number. To compute 5² , we multiply 5 two times (5×5) , and to compute 5³ , we multiply 5 three times (5×5×5) .
Exponents can be greater than 2 or 3. In fact, an exponent can be any number. We write an expression such as " 7⁴ " and say "seven to the fourth power." Similarly, 5⁹ is "five to the ninth power," and 11⁵⁶ is "eleven to the fifty-sixth power."
Since any number times zero is zero, zero to any (positive) power is always zero. For example, 0³¹ = 0 .
A number to the first power is that number one time, or simply that number: for example, 6¹ = 6 and 53¹ = 53 . We define a number to the zero power as 1: 8⁰ = 1 , (- 17)⁰ = 1 , and 521⁰ = 1 .
Here is a list of the powers of two:

20	=	1
21	=	2
22	=	2×2 = 4
23	=	2×2×2 = 8
24	=	2×2×2×2 = 16
25	=	2×2×2×2×2 = 32

and so on...

Exponents and the Base Ten System

Here is a list of the powers of ten:

100	=	1
101	=	10
102	=	10×10 = 100
103	=	10×10×10 = 1, 000
104	=	10×10×10×10 = 10, 000
105	=	10×10×10×10×10 = 100, 000

and so on... Look familiar? 10⁰ is 1 one (a 1 in the ones place), 10¹ is 1 ten (a 1 in the tens place), 10² is 1 hundred, 10³ is 1 thousand, 10⁴ is 1 ten thousand, etc. This is the meaning of base ten--a "1" in each place represents a number in which the base is 10 and the exponent is the number of zeros after the 1. The place value is the number that is multiplied by this number. For example, a 5 in the thousands place is equivalent to 5×1000 , or 5×10³ .
We can write out any number as a sum of single-digit numbers times powers of ten. The number 492 has a 4 in the hundreds place (4×10²) , a 9 in the tens place (9×10¹) and a 2 in the ones place (2×10⁰) . Thus, 492 = 4×10² +9×10¹ +2×10⁰ .

Examples: Write out the following numbers as single-digit numbers times powers of ten.

935 = 9×10² +3×10¹ +5×10⁰
67, 128 = 6×10⁴ +7×10³ +1×10² +2×10¹ +8×10⁰
4, 040 = 4×10³ +0×10² +4×10¹ +0×10⁰ 

 Powers of Negative Numbers, Decimals, and Fractions 

 Powers of Negative Numbers

Since an exponent on a number indicates multiplication by that same number, an exponent on a negative number is simply the negative number multiplied by itself a certain number of times:

(- 4)³ = - 4× -4× - 4 = - 64
(- 4)³ = - 64 is negative because there are 3 negative signs
(- 5)² = - 5× - 5 = 25
(- 5)² = 25 is positive because there are 2 negative signs.
Since an odd number of negative numbers multiplied together is always a negative number and an even number of negative numbers multiplied together is always a positive number, a negative number with an odd exponent will always be negative and a negative number with an even exponent will always be positive. So, to take a power of a negative number, take the power of the (positive) opposite of the number, and add a negative sign if the exponent is odd.

Example 1: (- 3)⁴ = ?

1. Take the power of the positive opposite. 3⁴ = 81 .
2. The exponent (4) is even, so (- 3)⁴ = 81 .

Example 2: (- 7)³ = ?

1. Take the power of the positive opposite. 7³ = 343
2. The exponent (3) is odd, so (- 7)³ = - 343 .

Powers of Decimals

When we square 0.46, we must remember that we are multiplying 0.46×0.46 , not 0.46×46 . In other words, the result has 4 decimal places, not 2.

0.46² = 0.46×0.46 = 0.2116.

When taking the power of a decimal, first count the number of decimal places in the base number, as when multiplying decimals. Next, multiply that number by the exponent. This will be the total number of decimal places in the answer. Then, take the power of the base number with the decimal point removed. Finally, insert the decimal point at the correct place, calculated in the second step.

Example 1: 1.5⁴ = ?

1. There is 1 decimal place and the exponent is 4. 1×4 = 4 .
2. 15⁴ = 50, 625 .
3. Insert the decimal point 4 places to the right. 1.5⁴ = 5.0625 .

Example 2: 0.04³ = ?

1. There are 2 decimal places and the exponent is 3. 2×3 = 6 .
2. 4³ = 64 = 000064 .
3. Insert the decimal point 6 places to the right. 0.04³ = 0.000064 .

As we can see, decimals less than 1 with large exponents are generally very small.

Powers of Fractions

The meaning of (3/4)³ is (3/4)×(3/4)×(3/4) , or three-fourths of three-fourths of three-fourths.When we multiply fractions together, we multiply their numerators together and we multiply their denominators together. To evaluate (3/4)³ = (3/4)×(3/4)×(3/4) , we multiply 3×3×3 , or 3³ , to get the numerator and we multiply 4×4×4 , or 4³ , to get the denominator. Thus, (3/4)³ = (3³)/(4³) .
To take the power of a fraction, take the power of the numerator to get the numerator, and take the power of the denominator to get the denominator. To take the power of a mixed number, convert the mixed number into an improper fraction and then proceed as above.

Examples:

I. (5/2)⁴ = (5⁴)/(2⁴) = 625/16

II. (- 3/4)² = ((- 3)²)/(4²) = 9/16

III. (1/(- 7))³ = (1³)/((- 7)³) = 1/(- 343) = - 1/343

Negative Exponents 

 

Negative Exponents

Taking a number to a negative exponent does not necessarily yield a negative answer. Taking a base number to a negative exponent is equivalent to taking the base number to the positive opposite of the exponent (the exponent with the negative sign removed) and placing the result in the denominator of a fraction whose numerator is 1. For example, 5^-4 = 1/5⁴ = 1/625 . 6^-3 = 1/6³ = 1/216 , and (- 3)^-2 = 1/(- 3)² = 1/9 .
If the base number is a fraction, then the negative exponent switches the numerator and the denominator. For example, (2/3)^-4 = (3/2)⁴ = (3⁴)/(2⁴) = 81/16 and (- 5/6)^-3 = (6/(- 5))³ = (6³)/((- 5)³) = 216/(- 125) = - 216/125 .

Negative Exponents and the Base Ten System

Here is a list of negative powers of ten:

10-1	=	1/101 = 1/10 = 0.1
10-2	=	1/102 = 1/100 = 0.01
10-3	=	1/103 = 1/1, 000 = 0.001
10-4	=	1/104 = 1/10, 000 = 0.0001
10-5	=	1/105 = 1/100, 000 = 0.00001

and so on...
Just as 10² represents a 1 in the hundreds place, 10^-2 represents a 1 in the hundredths place. The single-digit number in the hundredths place is the number that is multiplied by 10^-2 .
Now we can write out any terminating decimal as a sum of single- digit numbers times powers of ten. The number 23.45 has a 2 in the tens place (2×10¹) , a 3 in the ones place (3×10⁰) , a 4 in the tenths place (4×10^-1) and a 5 in the hundredths place (5×10^-2) . Thus, 23.45 = 2×10¹ +3×10⁰ +4×10^-1 +5×10^-2 .

Examples: Write out the following numbers as single-digit numbers times powers of ten:

523.81 = 5×10² +2×10¹ +3×10⁰ +8×10^-1 +1×10^-2

3.072 = 3×10⁰ +0×10^-1 +7×10^-2 +2×10^-3

46.904 = 4×10¹ +6×10⁰ +9×10^-1 +0×10^-2 +4×10^-3

 Square Roots 

 

Square Roots

The square root of a number is the number that, when squared (multiplied by itself), is equal to the given number. For example, the square root of 16, denoted 16^1/2 or , is 4, because 4² = 4×4 = 16 . The square root of 121, denoted , is 11, because 11² = 121 . = 5/3 , because (5/3)² = 25/9 . = 9 , because 9² = 81 . To take the square root of a fraction, take the square root of the numerator and the square root of the denominator. The square root of a number is always positive.
All perfect squares have square roots that are whole numbers. All fractions that have a perfect square in both numerator and denominator have square roots that are rational numbers. For example, = 9/7 . All other positive numbers have squares that are non-terminating, non- repeating decimals, or irrational numbers. For example, = 1.41421356... and = 2.19503572....

Square Roots of Negative Numbers

Since a positive number multiplied by itself (a positive number) is always positive, and a negative number multiplied by itself (a negative number) is always positive, a number squared is always positive. Therefore, we cannot take the square root of a negative number.
Taking a square root is almost the inverse operation of taking a square. Squaring a positive number and then taking the square root of the result does not change the number: = = 6 . However, squaring a negative number and then taking the square root of the result is equivalent to taking the opposite of the negative number: = = 7 . Thus, we conclude that squaring any number and then taking the square root of the result is equivalent to taking the absolute value of the given number. For example, = | 6| = 6 , and = | - 7| = 7 .
Taking the square root first and then squaring the result yields a slightly different case. When we take the square root of a positive number and then square the result, the number does not change: ()² = 11² = 121 . However, we cannot take the square root of a negative number and then square the result, for the simple reason that it is impossible to take the square root of a negative number.

Cube Roots and Higher Order Roots

A cube root is a number that, when cubed, is equal to the given number. It is denoted with an exponent of "1/3". For example, the cube root of 27 is 27^1/3 = 3 . The cube root of 125/343 is (125/343)^1/3 = (125^1/3)/(343^1/3) = 25/7 .
Roots can also extend to a higher order than cube roots. The 4th root of a number is a number that, when taken to the fourth power, is equal to the given number. The 5th root of a number is a number that, when taken to the fifth power, is equal to the given number, and so on. The 4th root is denoted by an exponent of "1/4", the 5th root is denoted by an exponent of "1/5"; every root is denoted by an exponent with 1 in the numerator and the order of root in the denominator.
An odd root of a negative number is a negative number. We cannot take an even root of a negative number. For example, (- 27)^1/3 = - 3 , but (- 81)^1/4 does not exist.

Fractional Exponents

We have just learned that a fractional exponent with "1" in the numerator is a root of some sort. But what would an exponent of "2/3" mean? Or an exponent of " -5/2 "?
In a fractional exponent, the numerator is the power to which the number should be taken and the denominator is the root that should be taken. For example, 64^2/3 means "square 64 and take the cube root of the result" or "take the cube root of 64 and square the result. This works out to 16.
A negative fractional exponent works just like a negative exponent. First, we switch the numerator and the denominator of the base number, and then we apply the positive exponent. For example, (9/25)^-5/2 = (25/9)^5/2 = (25^5/2)/(9^5/2) = "the square root of 25 to the fifth power over the square root of 9 to the fifth power" = 3, 125/243 . 27^-1/3 = (1/27)^1/3 = (1^1/3)/(27^1/3) = 1/3 .
Again, we cannot take a negative number to a fractional power if the denominator of the exponent is even.

Simplifying and Approximating Roots

Simplifying Square Roots

Often, it becomes necessary to simplify a square root; that is, to remove all factors that are perfect squares from inside the square root sign and place their square roots outside the sign. This action ensures that the irrational number is the smallest number possible, making it is easier to work with. To simplify a square root, follow these steps:

1. Factor the number inside the square root sign.
2. If a factor appears twice, cross out both and write the factor one time to the left of the square root sign. If the factor appears three times, cross out two of the factors and write the factor outside the sign, and leave the third factor inside the sign. Note: If a factor appears 4, 6, 8, etc. times, this counts as 2, 3, and 4 pairs, respectively.
3. Multiply the numbers outside the sign. Multiply the numbers left inside the sign.
4. Check: The outside number squared times the inside number should equal the original number inside the square root.

To simplify the square root of a fraction, simplify the numerator and simplify the denominator.
Here are some examples to make the steps clearer:

Example 1: Simplify 12^1/2 .

1. =
2. = 2×
3. 2× = 2×
4. Check: 2²×3 = 12

Example 2: Simplify .

1. =
2. = 2×5×
3. 2×5× = 10×
4. Check: 10²×6 = 600

Example 3: Simplify .

1. =
   
2. = 3×3×
   
3. 3×3× = 9×
4. Check: 9²×10 = 810

Similarly, to simplify a cube root, factor the number inside the " (  )^1/3 " sign. If a factor appears three times, cross out all three and write the factor one time outside the cube root sign.

Approximating Square Roots

It is very difficult to know the square root of a number (other than a perfect square) just by looking at it. And one cannot simply divide by some given number every time to find a square root. Thus, is it helpful to have a method for approximating square roots. To employ this method, it is useful to first memorize the square roots of the perfect squares. Here are the steps to approximate a square root:

1. Pick a perfect square that is close to the given number. Take its square root.
2. Divide the original number by this result.
3. Take the arithmetic mean of the result of I and the result of II by adding the two numbers and dividing by 2 (this is also called "taking an average").
4. Divide the original number by the result of III.
5. Take the arithmetic mean of the result of III and the result of IV.
6. Repeat steps IV-VI using this new result, until the approximation is sufficiently close.

If the square root can be simplified, it is easier to simplify and then approximate the number inside the " (  )^1/2 " sign. This result can then be multiplied by the number outside the " (  )^1/2 " sign.

Examples

Here are some examples to make the steps clearer:

Example 1: Approximate .

1. 25 is close to 22. = 5
2. 22/5 = 4.4
3. (5 + 4.4)/2 = 4.7
4. 22/4.7 = 4.68
5. (4.7 + 4.68)/2 = 4.69
6. 22/4.69 = 4.69

= 4.69

Example 2: Approximate .

1. 71 is close to 64. = 8
2. 71/8 = 8.9
3. (8 + 8.9)/2 = 8.45
4. 71/8.45 = 8.40
5. (8.45 + 8.40)/2 = 8.425
6. 71/8.425 = 8.427
7. (8.425 + 8.427)/2 = 8.426
8. 71/8.426 = 8.426

= 8.426

Example 3: Approximate .
can be simplified: = = 2× = 2×
Approximate :

1. 14 is close to 16. = 4
2. 14/4 = 3.5
3. (4 + 3.5)/2 = 3.75
4. 14/3.75 = 3.73
5. (3.75 + 3.73)/2 = 3.74
6. 14/3.74 = 3.74

= 3.74

Thus, = 2× = 2×3.74 = 7.48 Note that the eventual result will be the same no matter what perfect square one picks in Step 1.

Operations with Exponents

Order of Operations with Exponents

 We learned the order of operations, which uses these steps:

Step 1. Carry out the operations within the parentheses.
Step 2. Multiply and divide (it does not matter which comes first).
Step 3. Add and subtract (it does not matter which comes first).

But where do calculations with exponents fit into the picture?
The new order of operations, including operations with exponents, is:

Step 1. Carry out the operations within the parentheses (or absolute value).
Step 2. Evaluate powers and roots.
Step 3. Multiply and divide (it does not matter which comes first)--this includes applying a minus sign.
Step 4. Add and subtract (it does not matter which comes first).

Note that taking a power always comes before applying a minus sign unless the sign is inside the parentheses which form the base number. Observe:

-7² = - 49
(- 7)² = 49
-2⁴ = - 16
(- 2)⁴ = 16
-5³ = - 125
(- 5)³ = - 125 (because 3 is odd)
- = - 4
is undefined (we cannot take the square root of a negative number)