Exponential Notation With Positive Exponents

Exponential Notation

Learning Objective(south)

· Evaluate expressions containing exponents.

· Evaluate exponential notations with exponents of 0 and 1.

· Write an exponential expression involving negative exponents with positive exponents.

Introduction

A common language is needed in lodge to communicate mathematical ideas conspicuously and efficiently. Exponential notation is i example. It was developed to write repeated multiplication more efficiently. For example, growth occurs in living organisms by the sectionalisation of cells. Ane type of cell divides two times in an hour. Then in 12 hours, the cell volition carve up two • 2 • ii • 2 • 2 • ii • 2 • 2 • 2 • 2 • 2 • 2 times. This tin be written more efficiently equally 2¹².

We use exponential notation to write repeated multiplication, such equally x • 10 • 10 equally 10³. The 10 in 10ⁱⁱⁱ is called the base of operations. The 3 in 10³ is called the exponent. The expression 10^three is called the exponential expression.

base →  ten^{3 ←exponent}

10ⁱⁱⁱ is read as "10 to the 3rd power" or "x cubed." It means 10 • 10 • ten, or 1,000.

8² is read as "viii to the second power" or "8 squared." Information technology ways 8 • 8, or 64.

5^four is read every bit "five to the 4th ability." It ways five • v • 5 • five, or 625.

b ⁵ is read as " b to the fifth ability." Information technology means b • b • b • b • b . Its value will depend on the value of b.

The exponent applies only to the number that it is next to. And so in the expression xy ⁴, only the y is affected past the 4. xy ⁴ means 10 • y • y • y • y.

If the exponential expression is negative, such as − three^iv, it means –(3 •  3 • 3 • 3) or − 81.

If − 3 is to exist the base, information technology must be written as ( − iii)⁴, which means − 3 • − 3 • − 3 • − iii, or 81.

As well, ( − x )⁴ = ( − x ) • ( − x ) • ( − x ) • ( − x )  = x ^iv, while − x⁴ = –(x • x • x • 10).

Y'all can meet that there is quite a difference, so you have to be very careful!

Evaluating Expressions Containing Exponents

Evaluating expressions containing exponents is the same equally evaluating any expression. Y'all substitute the value of the variable into the expression and simplify.

Yous tin use PEMDAS to remember the order in which you should evaluate the expression. Beginning, evaluate anything in Parentheses or grouping symbols. Side by side, look for Exponents, followed by Gultiplication and Division (reading from left to correct), and lastly, Addition and Subtraction (once more, reading from left to right).

And so, when you lot evaluate the expression 5ten ³ if ten = four, outset substitute the value 4 for the variable ten . Then evaluate, using club of operations.

Example
Problem	Evaluate. five10 three if x = 4
	5 • ivthree	Substitute four for the variable x.
	5(4 • 4 • four) = 5 • 64	Evaluate four3 .
	320	Multiply.
Answer	vten 3 = 320 when x = 4

Observe the divergence between the instance higher up and the one below.

Instance
Problem	Evaluate. (5ten)3 if ten = iv
	(five • 4)iii	Substitute 4 for the variable ten.
	xx3	Multiply.
	20 • 20 • 20 = eight,000	Evaluate 203.
Answer	(5ten)three = 8,000 when x = 4

The addition of parentheses made quite a departure!

Case
Trouble	Evaluate. xiii if x = − four
	( − iv)iii	Substitute − iv for the variable x.
	− iv · − 4 · − 4	Evaluate.
	− four · − four · − four = − 64	Multiply.
Answer	x 3 = − 64, when x = − 4

Evaluate the expression − (iix)⁴, if ten = 3.

A) 1,296

B) − ane,296

C) 162

D) − 162

Prove/Hibernate Answer

A) one,296

Wrong. Substitute the value of 3 for the variable x and evaluate –(2 • iii)⁴. Do not utilise the negative sign until after yous have evaluated the expression (six)^iv. The correct answer is − 1,296.

B) − 1,296

Correct. Substitute the value of 3 for the variable x and evaluate –(2 • iii)⁴ = –vi^iv = –1,296.

C) 162

Wrong. Substitute the value of three for the variable x and evaluate –(2 • 3)⁴. Utilize the exponent four to the product 2 • 3, or 6. And then utilise the negative sign. The correct answer is − ane,296.

D) − 162

Incorrect. Substitute the value of 3 for the variable x and evaluate –(2 • iii)⁴. Apply the exponent 4 to the product ii • 3, or 6. Then use the negative sign. The correct answer is − ane,296.

Exponents of Zero and I

What does it mean when an exponent is 0 or 1? Let'south consider 25¹ . Any value raised to the ability of 1 is just the value itself. This makes sense, considering the exponent of ane means the base is used as a factor simply once. So the base of operations stands lonely, and 25¹ is simply 25.

Merely what about a value raised to the power of 0? Use what you know about powers of 10 to notice out what the power of 0 means. Beneath is a list of powers of 10 and their equivalent values. Look at how the numbers change going down the left and correct columns. There's a pattern in that location—see information technology?

Exponential Form	Expanded Form	Value
ten5	10 • 10 • 10 • 10 • 10	100,000
10iv	10 • 10 • 10 • ten	10,000
103	x • ten • 10	ane,000
xtwo	ten • 10	100
x1	10	x

Moving down the table, each row drops one factor of 10 from the one above it. From row 1 to row 2, the exponential course goes from x⁵ to 10⁴. The value drops from 100,000 to ten,000. Another way to put this is that each value is divided by x to produce the next value down the column.

Let's utilise this pattern of partitioning past 10 to predict the value of 10⁰.

Exponential Form	Expanded Form	Value
xfive	10 • 10 • x • 10 • 10	100,000
104	10 • 10 • 10 • 10	10,000
103	10 • 10 • 10	1,000
10two	10 • 10	100
10i	10	10
100	1	1

Following the pattern, you see that 10⁰ is equal to 1. Would the pattern hold for a different base? Say a base of 3?

Exponential Form	Expanded Grade	Value
iii5	3 • 3 • 3 • 3 • 3	243
3four	3 • iii • 3 • three	81
3iii	3 • 3 • 3	27
iii2	3 • three	9
31	3	3
three0	1	1

Yeah! And the aforementioned pattern would hold truthful for any non-goose egg number or variable raised to a power of 0, n ⁰ = 1.

There is a conflict when the base of operations is 0. You know that 0³ = 0, 0ⁱⁱ = 0, and 0¹ = 0, so you lot would await 0⁰ to as well be equal to 0. However, the above pattern says that any base raised to the ability of 0 is 1, then this leads y'all to believe that 0⁰ = 1. Notice the competing patterns—0⁰ cannot be both 0 and i! In this case, mathematicians say that the value of 0⁰ is undefined. (And think that undefined is non the same as 0!)

Exponents of 0 or 1

Whatsoever number or variable raised to a power of 1 is the number itself. n ¹ = n

Any non-zip number or variable raised to a power of 0 is equal to i. n ⁰ = 1

The quantity 0⁰ is undefined.

Example
Trouble	Evaluate. 2x 0 if x = ix
	2 • 90	Substitute 9 for the variable x.
	2 • 1	Evaluate 90.
	2	Multiply.
Answer	2x 0 = 2, if 10 = nine

As done previously, to evaluate expressions containing exponents of 0 or 1, substitute the value of the variable into the expression and simplify.

Evaluate the expression 3x ⁰ – y ¹, if x = 12 and y = –6.

A) 42

B) − 3

C) 9

D) two

Testify/Hibernate Answer

A) 42

Incorrect. Substitute the value of 12 for the variable x and –6 for the variable y: 3 • 12⁰ – ( − 6)¹. Call up that 12⁰ = 1. The correct answer is nine.

B) − three

Incorrect. Substitute the value of 12 for the variable x and –6 for the variable y: 3 • 12⁰ – ( − 6)ⁱ. Remember that (–6)¹ = − six and –( − half-dozen) = half dozen. The right respond is ix.

C) 9

Correct.  iii • 12⁰ – ( − 6)¹ = iii • one – ( − half-dozen)^one = 3 • i + vi = 9.

D) ii

Incorrect. Substitute the value of 12 for the variable x and − 6 for the variable y: 3 • 12⁰ – ( − 6)¹ . Remember that a base of operations raised to the power of 1 is the base. The correct answer is ix.

Negative Exponents

What does it mean when an exponent is a negative integer? Allow'due south employ the powers of ten pattern from before to detect out. If you continue this blueprint to add some more rows, beyond 10⁰, y'all detect the following:

Exponential Class	Expanded Form	Value
10five	10 • 10 • 10 • 10 • ten	100,000
104	x • 10 • 10 • 10	10,000
10three	10 • 10 • ten	1,000
102	ten • 10	100
x1	x	10
x0	1	ane
10-one
10-2

Following the pattern, you run into that 10⁰ is equal to 1. Then you get into negative exponents: 10^-1 is equal to , and 10^-2 is the same every bit .

Following this pattern, a number with a negative exponent can be rewritten equally the reciprocal of the original number, with a positive exponent.

For case, 10^-iii = and x^-7= .

To see if these patterns hold truthful for numbers other than 10, check out this table with powers of iii.

Exponential Grade	Expanded Class	Value
3five	3 • iii • 3 • 3 • 3	243
iiifour	3 • 3 • three • 3	81
three3	three • 3 • 3	27
32	iii • 3	9
31	3	3
30	i	i
3 - 1
3 - 2

The numbers are dissimilar just the patterns are the same. Nosotros are now ready to country the definition of a negative exponent.

Negative Exponent

For whatsoever not-zero number due north and any integer 10, northward^-x= . For example, v^-two= .

Note that the definition above states that the base, n must be a "non-goose egg number."

Evaluate the expression (x ^–2) • (x ⁰) when ten = six.

A)

B)

C) 0

D) 36

Testify/Hibernate Answer

A)

Correct.  Substitute the value of 6 for the variable x and evaluate. .

B)

Incorrect. Substitute the value of half-dozen for the variable x and evaluate: half dozen^-2 • 6⁰. Recall that (6)⁰ = 1. The correct respond is .

C) 0

Wrong. Substitute the value of 6 for the variable x and evaluate: 6^-2 • 6⁰. Remember that (six)⁰ = 1. The right answer is .

D) 36

Incorrect. Substitute the value of half dozen for the variable x and evaluate: 6^-ii • 6⁰. Remember that ( . The correct respond is .

Summary

Exponential notation is equanimous of a base of operations and an exponent. Information technology is a "shorthand" fashion of writing repeated multiplication, and indicates that the base is a factor and the exponent is the number of times the factor is used in the multiplication. The basic rules of exponents are as follows:

· An exponent applies but to the value to its firsthand left.

· When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.

· For any non-zippo number northward, n ⁰ = i.

· For any non-zero number due north and whatever integer 10, n ^–x = .

Exponential Notation With Positive Exponents,

Source: http://content.nroc.org/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U11_L1_T1_text_final.html

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