Learning Objectives

review the rule of exponents. Testimonial the definition of an adverse exponents and also zero together an exponent. Work with number using clinical notation.

Review the the rule of Exponents

In this section, we testimonial the rules of exponents. Recall the if a factor is repeated multiple times, then the product can be written in exponential kind xn. The positive integer exponent n suggests the variety of times the base x is repetitive as a factor.

You are watching: The rule governing exponents is that an exponent in the denominator is the same as a


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Expanding the expression making use of the an interpretation produces multiple components of the base which is fairly cumbersome, specifically when n is large. Because that this reason, we have useful rules to aid us leveling expressions v exponents. In this example, an alert that we could acquire the same result by adding the exponents.

x4⋅x6=x4+6=x10Product rule for exponents

In general, this explains the product preeminence for exponentsxm⋅xn=xm+n; the product of 2 expressions with the same base have the right to be simplified by adding the exponents.. In other words, as soon as multiplying two expressions v the same base we add the exponents. To compare this to raising a aspect involving an exponent come a power, such as (x6)4.


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Here we have 4 components of x6, i beg your pardon is equivalent to multiply the exponents.

(x6)4=x6⋅4=x24Power rule for exponents

This describes the power dominion for exponents(xm)n=xmn; a power raised to a power deserve to be simplified by multiplying the exponents.. Now we think about raising grouped assets to a power. For example,

(x2y3)4=x2y3 ⋅ x2y3 ⋅ x2y3 ⋅ x2y3=x2⋅x2⋅x2⋅x2 ⋅ y3⋅y3⋅y3⋅y3 Commutative property=x2+2+2+2⋅y3+3+3+3=x8y12

After expanding, we space left with four factors of the product x2y3. This is tantamount to raising each the the initial grouped components to the 4th power and applying the power rule.

(x2y3)4=(x2)4(y3)4=x8y12

In general, this defines the use of the power ascendancy for a product and the power preeminence for exponents. In summary, the rule of index number streamline the procedure of working through algebraic expressions and also will it is in used generally as we move through our research of algebra. Given any type of positive integers m and n where x,y≠0 we have


Product dominance for exponents:

xm⋅xn=xm+n

Quotient rule for exponents:

xmxn=xm−n

Power dominion for exponents:

(xm)n=xm⋅n

Power dominion for a product:

(xy)n=xnyn

Power preeminence for a quotient:

(xy)n=xnyn 


(xy)n=xnyn; if a product is raised to a power, then apply that power to each variable in the product.

(xy)n=xnyn ; if a quotient is raised to a power, then use that power to the numerator and the denominator.

These rules enable us to successfully perform operations through exponents.


Example 1

Simplify: 104⋅1012103.

Solution:

104⋅1012103=1016103 Product rule=1016−3 Quotient rule=1013

Answer: 1013


In the ahead example, notification that we did no multiply the basic 10 times itself. When using the product rule, add the exponents and also leave the base unchanged.


Example 2

Simplify: (x5⋅x4⋅x)2.

Solution:

Recall the the variable x is assumed to have an exponent that one, x=x1.

(x5⋅x4⋅x)2=(x5+4+1)2=(x10)2=x10⋅2=x20

Answer: x20


Example 3

Simplify: (x+y)9 (x+y)13.

Solution:

Treat the expression (x+y) as the base.

(x+y)9 (x+y)13=(x+y)9+13=(x+y)22

Answer: (x+y)22


The commutative building of multiplication allows us to use the product rule for index number to simplify determinants of an algebraic expression.


Example 4

Simplify: −8x5y  ⋅  3x7y3.

Solution:

Multiply the coefficients and add the index number of variable components with the same base.

−8x5y  ⋅  3x7y3=−8⋅3  ⋅ x5⋅x7  ⋅  y1⋅y3 Commutative property=  −24 ⋅ x5+7  ⋅  y1+3 Power rule for exponents=−24x12y4

Answer: −24x12y4


Example 5

Simplify: 33x7y5(x−y)1011x6y(x−y)3.

Solution:

33x7y5(x−y)1011x6y(x−y)3=3311  ⋅  x7−6 ⋅y5−1⋅(x−y)10−3=3x1y4 (x−y)7

Answer: 3xy4 (x−y)7


The power rule for a quotient permits us to use that exponent come the numerator and denominator. This dominance requires that the denominator is nonzero and also so we will make this assumption for the remainder that the section.


Example 6

Simplify: (−4a2bc4)3.

Solution:

First apply the power preeminence for a quotient and then the power dominance for a product.

(−4a2bc4)3=(−4a2b)3(c4)3 Power rule for a quotient=(−4)3(a2)3(b)3(c4)3 Power rule for a product=−64a6b3c12

Answer: −64a6b3c12


Using the quotient dominion for exponents, we can specify what it method to have zero together an exponent. Take into consideration the adhering to calculation:

1=2525=5252=52−2=50

Twenty-five divided by twenty-five is clearly equal to one, and when the quotient dominance for index number is applied, we see that a zero exponent results. In general, given any kind of nonzero genuine number x and also integer n,

1=xnxn=xn−n=x0

This leads united state to the meaning of zero together an exponentx0=1 ; any kind of nonzero base increased to the 0 strength is defined to be 1.,

x0=1  x≠0

It is essential to note that 00 is indeterminate. If the basic is negative, then the an outcome is still optimistic one. In various other words, any kind of nonzero base raised to the zero strength is defined to be equal to one. In the following examples assume all variables room nonzero.


Example 7

Simplify:

(−2x)0 −2x0

Solution:

Any nonzero amount raised come the zero strength is equal to 1.

(−2x)0=1

In the example, −2x0, the base is x, not −2x.

−2x0=−2⋅x0=−2⋅1=−2


Noting that 20=1 we deserve to write,

123=2023=20−3=2−3

In general, given any nonzero real number x and also integer n,

1xn=x0xn=x0−n=x−n  x≠0

This leads us to the definition of an adverse exponentsx−n=1xn , given any type of integer n, whereby x is nonzero.:

x−n=1xn  x≠0

An expression is completely simplified if the does no contain any an adverse exponents.


Example 8

Simplify: (−4x2y)−2.

Solution:

Rewrite the entire quantity in the denominator v an exponent the 2 and also then leveling further.

(−4x2y)−2=1(−4x2y)2=1(−4)2 (x2)2 (y)2=116x4y2

Answer: 116x4y2


The previous example suggests a property of quotients with an adverse exponentsx−ny−m=ymxn , given any type of integers m and n, where x≠0  and y≠0.. Given any integers m and also n wherein x≠0  and y≠0, then

x−ny−m= 1xn 1ym=1xn⋅ym1=ymxn

This leads us to the property

x−ny−m=ymxn 

In various other words, an adverse exponents in the numerator have the right to be created as hopeful exponents in the denominator and an unfavorable exponents in the denominator deserve to be composed as confident exponents in the numerator.


Example 10

Simplify: −5x−3y3z−4.

Solution:

Take care with the coefficient −5, acknowledge that this is the base and also that the exponent is actually positive one: −5=(−5)1. Hence, the rule of negative exponents do not use to this coefficient; leave it in the numerator.

−5x−3y3z−4=−5  x−3  y3z−4=−5 y3 z4x3

Answer: −5y3z4x3


Furthermore, all of the rules of exponents identified so far expand to any type of integer exponents. We will broaden the limit of these properties to include any real number exponents later on in the course.


Scientific Notation

Real number expressed using scientific notationReal number expressed the kind a×10n, where n is an integer and 1≤a10. Have actually the form,a×10nwhere n is one integer and also 1≤a10. This kind is specifically useful when the numbers space very large or an extremely small. Because that example,

9,460,000,000,000,000 m=9.46×1015 m One light year0.000000000025 m=2.5×10−11 m          Radius of a hydrogen atom

It is cumbersome to create all the zeros in both of this cases. Scientific notation is one alternative, compact depiction of this numbers. The variable 10n suggests the strength of ten to main point the coefficient by come convert ago to decimal form:


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This is tantamount to moving the decimal in the coefficient fifteen places to the right.

A an unfavorable exponent indicates that the number is really small:


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This is indistinguishable to moving the decimal in the coefficient eleven locations to the left.

 

Converting a decimal number to clinical notation entails moving the decimal together well. Consider every one of the equivalent develops of 0.00563 with factors of 10 that follow:

0.00563=0.0563×10−1=0.563×10−2=5.63 ×10−3=56.3×10−4=563×10−5

While every one of these space equal, 5.63×10−3 is the only kind expressed in correct clinical notation. This is since the coefficient 5.63 is between 1 and 10 as forced by the definition. An alert that we can convert 5.63×10−3 ago to decimal form, as a check, by moving the decimal three locations to the left.

xmxn=xm−n; the quotient of two expressions v the same base can be simplified by individually the exponents.


Example 11

Write 1,075,000,000,000 using clinical notation.

Solution:

Here us count twelve decimal areas to the left the the decimal allude to obtain the number 1.075.

1,075,000,000,000=1.075×1012

Answer: 1.075×1012


Example 12

Write 0.000003045 using clinical notation.

Solution:

Here we count six decimal locations to the ideal to achieve 3.045.

0.000003045=3.045×10−6

Answer: 3.045×10−6


Often us will have to perform operations when using numbers in clinical notation. Every the rules of exponents occurred so far also apply to number in clinical notation.


Example 13

Multiply: (4.36×10−5)(5.3×1012).

Solution:

Use the truth that multiplication is commutative, and apply the product rule for exponents.

(4.36×10−5)(5.30×1012)= (4.36⋅5.30)×(10−5⋅1012)=23.108×10−5+12=2.3108×101 ×  107=2.3108×101+7=2.3108×108

Answer: 2.3108×108


Example 14

Divide: (3.24×108)÷(9.0×10−3).

Solution:

(3.24×108)(9.0×10−3)=(3.249.0)×(10810−3)=0.36×108−(−3)=0.36×108+3=3.6×10−1×1011=3.6×10−1+11=3.6×1010

Answer: 3.6×1010


Example 15

The speed of light is around 6.7×108 miles every hour. Refer this speed in miles per second.

Solution:

A unit evaluation indicates that we need to divide the number by 3,600.

6.7×108 miles per hour=6.7×108 miles1  hour⋅(1  hour60 minutes)⋅(1 minutes60 seconds)=6.7×108 miles3600 seconds=(6.73600)×108  ≈0.0019×108 rounded to two significant digits=1.9×10−3×108 =1.9×10−3+8=1.9×105

Answer: The speed of irradiate is approximately 1.9×105 miles per second.


Example 16

The sun moves approximately the center of the galaxy in a virtually circular orbit. The street from the center of our galaxy to the sunlight is about 26,000 light-years. What is the circumference of the orbit the the Sun around the galaxy in meters?

Solution:

One light-year measures 9.46×1015 meters. Therefore, main point this by 26,000 or 2.60×104 to discover the length of 26,000 light years in meters.

(9.46×1015)(2.60×104)=9.46⋅2.60×1015⋅104≈24.6×1019=2.46×101⋅1019=2.46×1020

The radius r the this very huge circle is around 2.46×1020 meters. Usage the formula C=2πr to calculate the one of the orbit.

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C=2πr≈2(3.14)(2.46×1020)=15.4×1020=1.54×101⋅1020=1.54×1021

Answer: The one of the Sun’s orbit is roughly 1.54×1021 meters.


Key Takeaways

when multiplying two amounts with the exact same base, include exponents: xm⋅xn=xm+n. When splitting two amounts with the very same base, subtract exponents: xmxn=xm−n. When raising strength to powers, main point exponents: (xm)n=xm⋅n. When a grouped quantity involving multiplication and department is increased to a power, use that power to all of the determinants in the numerator and also the denominator: (xy)n=xnyn and also (xy)n=xnyn . Any kind of nonzero quantity raised come the 0 power is characterized to be equal to 1: x0=1 . Expressions with an adverse exponents in the numerator have the right to be rewritten together expressions with hopeful exponents in the denominator: x−n=1xn . Expression with negative exponents in the denominator can be rewritten as expressions with confident exponents in the numerator: 1x−m=xm . Take care to distinguish negative coefficients from an adverse exponents. Clinical notation is an especially useful once working through numbers that are very big or really small.

Topic Exercises

Part A: rules of Exponents

Simplify. (Assume every variables stand for nonzero numbers.)