exponential expressions

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Exponential expressions on the SAT test

Exponential expressions on the SAT test

SAT Subscore: Passport to advanced mathematics

Studying exponential expressions

On the SAT test exponential expressions are part of passport to advanced mathematics subscore that includes 9 advanced topics (see the full topics list on the top menu). 

Exponential expressions topic is the fourth topic of passport to advances mathematics subscore. It is recommended to start learning passport to advances mathematics subscore with its first three topics that deal with quadratic eqiations and functions (the first topic is quadratic equations and quadratic functions). 

Exponential expressions topic is divided into sections from easy to difficult (the list of the sections appears on the left menu). Each section includes detailed explanations of the required material with examples followed by a variety of self-practice questions with solutions.

Finish studying heart of algebra subscore topics before you study this topic or any other passport to advances mathematics subscore topic. (Heart of algebra subscore includes basic algebra topics which knowledge is required for understanding passport to advanced mathematics subscore topics). 

Exponential expressions- summary

Exponential expression includes 3 components: coefficient, base and exponent. For example: 5x3.
The exponent is the number of times we multiply x (in the example the exponent is 3, since x3=x*x*x).
The base is the variable (in the example the base is x).
The coefficient is the number multiplied by the variable (in the example the coefficient is 5, since 5x3=5*x3.

To solve an exponential expressions questions we need to calculate the values of exponential expressions or solve equations with exponents and radicals.

The skills required for exponential expressions topic include dealing with different types of exponents: positive exponents, zero exponents and rational exponents (exponents with fractions) and dealing with radicals (roots). You also need to know how to solve word problems with exponential expressions: population growth and decay problems and compounding interest problems.

Performing operations with exponential expressions requires:
Changing the base of exponential expression.
Raising an exponential expression to an exponent.
Adding, subtracting multiplying and dividing exponential expressions.
Multiplying polynomial expressions with the FOIL formula.

Performing operations with radicals (roots) requires:
Rewriting radicals as exponents.
Making calculations with exponents and radicals.
Multiplying and dividing radicals.

Solving word problems questions requires:
Writing an exponential expression from a word problem.
Solving a given exponential expression according to a word problem.
Rounding answers to the nearest whole number.
Calculating the net value of the change in the population.
Calculating the interest that is compounded between 2 periods.
Changing the time units (changing the exponent of the expression).

The formula sheet for rational exponents and radicals is listed below.

Exponential expressions formula sheet

The exponential expressions formula sheet is given below, it includes five types of formulas:

  1. Negative and zero exponents formulas.
  2. Formulas for raising an exponential expression to an exponent.
  3. Formulas for adding and subtracting exponential expressions.
  4. Formulas for multiplying and dividing exponential expressions.
  5. Formulas for exponent operations with radicals.

Note that all the formulas are explained in detail on this page. 

The types of exponential expressions

Exponential expression includes 3 components: coefficient, base and exponent. For example: 5x3.
The exponent is the number of times we multiply x (in the example the exponent is 3, since x3=x*x*x).
The base is the variable (in the example the base is x).
The coefficient is the number multiplied by the variable (in the example the coefficient is 5, since 5x3=5*x3.

what are the components of an exponential expression
  1. Positive exponents are exponents with positive numbers. For example: x5.
  2. Zero exponents have 0 as the exponent and they all are equal to 1. For example: x0=1000=1.
  3. Negative exponents are exponents with negative numbers. For example: x-3.
  4. Rational exponents or radicals.
What are the 4 types of exponents in the exponential expressions?

What are Monomial and Polynomial expressions?

Monomial is a term of a form axn for constant a and none-negative integer n (the meaning of the word “mono” is one).
Note that the power can be zero so that the monomial will be equal to a constant a. Since x0=1 we know that ax0=a*1=a. For example, 8x0=8*1=8.

Consider the following examples:
5x7 is a monomial because 5 is constant and 7 is a non-negative integer.
4x is not a monomial because x is a variable (and not a non-negative integer).
X1/4 is not a monomial because 1/4 is a fraction (and not a non-negative integer).

Polynomial expression is composed of different monomials (the meaning of the word “poly” is many).
Binomial is a polynomial with 2 monomials, for example 5x7-5.
Trinomial is a polynomial with 3 monomials, for example 5x7+x4-5

Negative and zero exponents

Negative exponents formula:
x-n=1/xn

For example: x-3=1/x3

Zero exponent formula:
x0=1, x≠0

For example: 1000=1

The rule for negative exponents:

A positive base with a negative exponent is equal to 1 divided by the base raised to the opposite of the exponent. Meaning that in order to change the sign of the base from negative to positive, we must move the base between the numerator and the denominator. The formula for a negative exponent is x-n=1/xn
For example: x-3=1/x3

Consider this example:

x=a-n/4, n>0

What is a equal to in terms of x?
(a=?)

First, we will rewrite the expression making the negative exponent positive using the formula x-n=1/xn
x=a-n/4
x=1/an/4

Multiply both sides by an/4
an/4x=1*an/4/an/4
an/4x=1

Divide both sides by x
an/4=1/x

In order to cancel the exponent of a we raise both sides by 4/n
a=1/x4/n

Another way to solve the question is to raise both sides of the equation by the opposite exponent to -n/4 which is -4/n (since (-n/4)*(-4/n)=1).
x=a-n/4
x-4/n=a

We can also write this as
a=1/x4/n

 

The rule for an exponent that is equal to zero:

Every base that is raised to an exponent of 0 equals to 1. The formula for a zero exponent is x0=1

X0=1, 1000=0……

Note that the base itself can’t be 0 (x≠0).

Changing the base of an exponential expression

We can write the same exponential expression with different bases. In order to find the equal expressions with different bases we need to check if we can make the base smaller or bigger.

1.Making the base smaller– Does the base have a square root or a cube root?
For example: 92=(32)2=34  (We changed the base from 9 to 3).
For example: 8=81=23 (We changed the base from 8 to 2).

2.Making the base bigger– Do we have an exponent in the expression?
For example: 92=811=81 (We changed the base from 9 to 81).

3.Making the base bigger– this is always possible. We can raise the base to an exponent and make a root to inverse the operation.
For example: 8=√64 or 641/2 (We changed the base from 8 to 64)

Consider this example:

If a2/b=16 for a and b that are positive integers, what are the values of a and b?

If we want the base on the left side of the equation to be 16 then we need to solve 162/b=161
2/b=1
b=2
a=16

Another option is to change the base of 16 to 4 by writing 16 as 42. Now we need to solve 4 2/b=42
2/b=2
b=1
a=4

Another option is to square the base of 16 to 256 by writing 16 as 2561/2. We need to solve 2562/b=2561/2
2/b=1/2
b=4
a=256

The rules for raising an exponential expression to an exponent

In these questions we have an exponential expression and we need to raise it to another exponent. In order to simplify the expression, we need to open the brackets by raising everything that is in the brackets to the exponent.

Formula- the base is constant:

(ab) n=abn

For example: (34) 2=34*2=38

Formula- The base includes a variable:

(axm) n=anxmn

For example: (2x2) 3=23x2*3=8x6

When the base is a constant (no variable) the formula is (ab) n=abn
For example: (34) 2=32*4=38
We can solve without using the formula: (34) 2=(3*3*3*3)*(3*3*3*3)=38

When the base also includes a variable, the formula is (axm) n=anxmn with 3 steps:

  1. Raise the coefficient to the exponent
  2. Keep the base
  3. Multiply the exponents

For example: (2x2) 3=23x2*3=8x6
We can solve without using the formula: (2x2) 3=2x2*2x2*2x2=8*x*x*x*x*x*x=8x6

The rules for adding and subtracting exponential expressions

We can add or subtract only exponential expressions that have the same base and the same exponent. This is done by combining like terms using the formula axn±bxn=(a±b)xn

Adding exponential expressions formula:

axn+bxn=(a+b)xn

For example: 3x3+2x3= (3+2)x3=5x3

Subtracting exponential expressions formula:

axn-bxn=(a-b)xn

For example: 3x3-2x3= (3-2)x3=x3

If we add or subtract expressions that have the same base and the same exponent, we need to use the formula axn±bxn=(a±b)xn with 3 steps:

  1. Keep the base.
  2. Keep the exponent.
  3. Add or subtract the coefficients.

Note that if the bases are different (3x3-2y3) or the exponents are different (3x5-2y3), we can’t add or subtract the expressions.

For example: 3x3-2x3=?
In both of the expressions the base is x and the exponent is 3 therefore we can use the formula.
3x3-2x3= (3-2)x3=x3
We can also solve the expression as 3x3-2x3= x3+x3+x3-x3-x3= x3

 

Adding and subtracting polynomials with different exponents and different bases:

If we add or subtract polynomials with different exponents and different bases we first need to group like terms and then follow the steps listed above.

  1. Group like terms
  2. Keep the bases
  3. Keep the exponents
  4. For each group off equal bases and equal exponents add or subtract the coefficients.

Consider this example:

Solve 8x3+2x-5x3-(5-4x)

Open brackets:
8x3+2x-5x3-5- -4x=8x3+2x-5x3-5+4x

Group like terms (find equal bases and equal exponents):
8x3-5x3+4x+2x-5

For each group off equal bases and equal exponents add or subtract the coefficients with the formula axn±bxn=(a±b)xn:
(8-5)x3+(2+4)x-5
group 1 includes expressions with x3, group 2 includes expressions with x1 and group 3 includes expressions with x0

Continue solving: 3x3+6x-5

The rules for multiplying and dividing exponential expressions

We can multiply or divide only exponential expressions that have the same base or the same exponent.

Multiplying and dividing exponential expressions with the same base:

Formula for myltiplying exponential expressions with same base:
axm*bxn=(a*b)xm+n

For example: 2x2 *4x3=8x5

Formula for dividing exponential expressions with the same base:
axm/bxn=a/b*xm-n

For example: 6x5 /2x3=3x2

If we multiply expressions that have the same base, we need to use the formula axm*bxn=(a*b)xm+n with 3 steps:

  1. Multiply the coefficients
  2. Keep the base
  3. Add the exponents

For example: 2x2 *4x3=(2*4)x2+3 =8x5

We can also solve the expression without using the formula:  2x2 *4x3=2*4*x*x*x*x*x=8*x5=8x5

If we divide expressions that have the same base, we need to use the formula axm/bxn =a/b*xm-n with 3 steps:

  1. Divide the coefficients
  2. Keep the base
  3. Subtract the exponents

For example: 6x5 /2x3=(6/2)x5-3 =3x2

We can also solve the expression without using the formula:  6x5 /2x3=6/2*(x*x*x*x*x )/(x*x*x)=3*x*x =3x2

Multiplying exponential expressions with the same exponents:

Formula for myltiplying exponential expressions with the same exponents: 

axn*byn=ab(xy)n

For example: 6x3 *2y3=12(xy)3

Formula for dividing exponential expressions with the same exponents:

(axn)/(byn)=(a/b)(x/y)n

For example: 6x3 /2y3=3(x/y)3

If we multiply expressions that have the same exponent, we need to use the formula axn*byn=ab(xy)n with 3 steps:

  1. Multiply the coefficients
  2. Multiply the bases
  3. Keep the exponent

For example: 6x3 *2y3=6*2*x3*y3 =12(xy)3

If we divide expressions that have the same exponent, we need to use the formula (axn)/(byn)=(a/b)(x/y)n with 3 steps:

  1. Divide the coefficients
  2. Divide the bases
  3. Keep the exponent

For example: 6x3 /2y3=(6/2)*(x3/y3)=3(x/y)3

Note that if the bases are different and the exponents are different (3x5*2y3 or 4x5/2y3), we can’t multiply or divide the expressions. The only thing we can do is to multiply or divide the coefficients.

Note that if the bases and the exponents are the same, we can solve the expression with each one of the formulas.
For example:
2x3*4x3=2*4*x3+3=8x
2x3*4x3=2*4*(x*x) 3 =8*(x2) 3 =8x6

The rules for multiplying polynomial expressions

FOIL formula for multiplying two binomials:
(ax+b)(cx+d)=ax*cx+ax*d+b*cx+b*d

For example: (2x+5)(3x-3)=2x*3x+2x*-3+5*3x+5*-3

Monomial is a term of a form axn for constant a and none- negative integer n. For example: 5x7.

Polynomial expression is composed of different monomials. For example: 5x7+x4-5.

Multiplying monomial and a polynomial:

Multiplying monomial  and a polynomial requires us to write the polynomial in brackets and then distribute by opening brackets. Then we need to continue solving the distributed terms according to the exponent rules explained above.

Assume that we need to multiply a monomial and a trinomial, for example multiply 3 and 5x6-4x-2.  We write the multiplication as 3(5x6-4x-2) and then solve by multiplying 3 by every number inside the brackets.

3(5x6-4x-2)=3*5x6+3*-4x+3*-2=15x6-12x-6

Multiplying two binomials with FOIL method:

Multiplying two binomials requires us to write each binomial in brackets and then distribute by opening brackets using the FOIL formula. Then we need to continue solving the distributed terms according to the exponent rules explained above.

Assume that we need to multiply 2 binomials ax+b and cx+d, in this case we need to write each binomial in brackets and then solve. Since we have 2 components in each binomial, we know that the number of products in the solution will be 2*2=4.

The FOIL formula for multiplying 2 binomials is (ax+b)(cx+d)=ax*cx+ax*d+b*cx+b*d

The letters FOIL stand for First, Outer, Inner, Last:

First- Multiply the first terms ax*cx

Outer- Multiply the outer terms ax*d

Inner- Multiply the inner terms b*cx

Last- Multiply the outer terms b*d

Note that if one of the numbers is negative we must keep it’s negative sign when opening brackets.

Consider this example:

What is the product of 2x+5 and 3x-3?

Plugging into FOIL formula we will get:

First: 2x*3x

Outer: 2x*-3

Inner: 5*3x

Last: 5*-3

The 4 parts of the formula are: (2x+5)(3x-3)=2x*3x+2x*-3+5*3x+5*-3

The next step is to continue solving the distributed terms according to the exponent rules explained above:

2x*3x+2x*-3+5*3x+5*-3=6x2-6x+15x-15=6x2+9x-15

Rational exponents

Rational exponents are exponents that are fractions. For example X1/2.

All the rules that apply to integer exponents also apply to rational exponents.

Raising an exponential expression to an exponent: (axm) n=anxmn
For example: (4x1/2)1/2=41/2x1/2*1/2=2x1/4

Multiply expressions that have the same base: axm*bxn=(a*b)xm+n
For example: 4x8/3*5x4/3=4*5*x8/3+4/3=20x12/3=20x4

Divide expressions that have the same base: axm/bxn=a/b*xm-n
For example: 15x7/3/3x1/3=(15/3)*(x) 7/3-1/3=5x6/3=5x2

Multiply expressions that have the same exponent: axn*byn=ab(xy)n
For example: 2x1/3*3y1/3=2*3*(xy)1/3=6(xy)1/3

Divide expressions that have the same exponent: (axn)/(byn)=(a/b)(x/y)n
For example: (4x1/6)/(2y1/6)=(4/2)(x/y)1/6=2(x/y)1/6

Radicals

Radicals are rational exponents that are written with roots. For example √x. The symbol of a radical is √ and it represents a square root (instead of writing 2√x we write only √x).

The connection between rational exponents and radicals:

Formula for rewriting radicals and exponents:
n√x m =xm/n

For example: x3/4=4√x3

Rational exponents are exponents that are fractions and radicals are rational exponents that are written with roots. Therefore, radicals are just another way to present rational exponents.

When rewriting a radical as a fractional exponent, the exponent under the radical symbol becomes the numerator and the value to the left of the radical symbol becomes the denominator. The formula is n√x m =xm/n

For example:
n√x1=x1/n
x1/2=√x
x1/5=5√x
x3/4=4√x3

Consider this example:

Write the expression x1/5y3/4 as radical.

In order to solve the example, we need to use the formula for rewriting fractional exponents as radicals: xm/n= n√x m
X1/5= 5√x 1= 5√x
y3/4= 4√y 3
x1/5y3/4=5√x 4√y 3   

The connection between exponents and radicals:

Squares and square roots are inverse operations. In order to see this, we need to rewrite the square root as a fractional exponent.
(√x) 2= (x1/2)2=x1/2*2=x

 

We can also see this with different roots:
(3√x) 3= (x1/3)3=x1/3*3=x

The rules of exponent operations with radicals:

Formula for myltiplying radicals

n√x* n√y = n√(xy)

For example: 4√x* 4√y = 4√(xy)

Formula for dividing radicals:

n√x/ n√y = n√(x/y)

For example: 3√x/ 3√y = 3√(x/y)

All rules that apply to integer exponents also apply to radicals.
Multiply expressions that have the same exponent: n√x* n√y = n√(xy)
For example: 4√x* 4√y = 4√(xy)

Divide expressions that have the same exponent: n√x/ n√y = n√(x/y)
For example: 3√x/ 3√y = 3√(x/y)

Consider this example:

Solve the expression √16x8y4/ 3√x6y3

First, we will write the radicals as exponential fractions
√16x8y4/ 3√x6y3=(16x8y4)1/2/(x6y3)1/3

Open brackets using the formula (axm) n=anxmn
(16x8y4)1/2/(x6y3)1/3=(161/2x8/2y4/2)/(x6/3y3/3)=(4x4y2)/(x2y)

Cancel the denominator using the formula axm/bxn=a/b*xm-n
(4x4y2)/(x2y)=4x4-2y2-1=4x2y

Word problems with exponential expressions

There are 2 main types of word problems with exponential functions:  population growth and decay problems and compounding interest problems. The function that is modeling these word problems is f(t)=a(b),where a reprsents the inial amount, b represents the change in the inital amount and t represents time units.

The skills required are:
Writing an exponential function from a word problem.
Solving a given exponential function according to a word problem.
Rounding answers to the nearest whole number.
Calculating the net value of the change in the population.
Calculating the interest that is compounded between 2 periouds.
Changing the time units (changing the exponent of the expression).

Population growth and decay word problems

The function that is modeling population grows and decay is p(t)=p0(r)with the following components:
p(t) is the output that represents the population.
p0– represents the initial amount of the population.
t- represents time (usually in years).
r- represents the change in the population (for t measured in years, each year the population is r times the population in the previous year).

Note that:
1. If r>1 then the population is growing, if 1>r>0 then the population is declining.
2. Since the population is a whole number you will need to round down the answer to the nearest whole number. Note that you must round down and not up.
3. If you are asked about the net increase of the population, you need to subtract the initial amount of the population from the population you calculate with the formula, meaning the net increase is p(t)-p0.

Consider this example:

The population of Z town is declining at the rate of 2% per year.

If the population of Z town is 5400 people, what is the change in the population each year in the first 3 years?

In this question we need to write the function that models the population. Let’s look at the population amounts each year: the population every year is the population in the previous year multiplies by 0.98.

Note that the decrease in the population is not constant. The reason is that the decrease is calculated as 0.98 multiplied by the population in the previous year and the population each year us different.

The population in t=0 is 5400

The population in t=1 is:
5400*(1-0.02)1=5400*0.98=5292
5400-0.02*5400=5292
(the decrease is 5292-5400=-108 that is 5400*0.02)

The population in t=2 is:
5400*(1-0.02)2=5400*0.98*0.98=5186
5400-0.02*5400-0.02*5292=5292-0.02*5292=5186
(the decrease is 5186-5292=-106 that is 5292*0.02)

The population in t=3 is
5400*(1-0.02)3=5400*0.98*0.98*0.98=5082
5400-0.02*5400-0.02*5292-0.02*5186=5186-0.02*5186=5082
(the decrease is 5082-5186=-104 that is 5186*0.02)

Consider this example:

The number of bacteria in a population after t minutes is modeled by a function p(t)=100(1.2)t.

What is the number of the bacteria after 2 minutes?

p(t)=100(1.2)t

p(t=3)=100(1.2)2=100*1.2*1.2=100*1.44=144

The number of bacteria after 2 minutes is 144.

Consider this example:

The number of bacteria in a population after t minutes is modeled by a function p(t)=100(1.2)t.

What is the net increase of bacteria population after 5 minutes? Round your answer to the nearest whole number.

p(t)=100(1.2)t

p(t=2)=100(1.2)5=100*2.49=248.832=248 rounded

The net increase is 248-100=148 bacteria.

Consider this example:

p(t)=3050(1.05)t

The function above models the population of a town C, t years after the year 2018. What is the net increase of the town’s population from 2018 to 2022 ? Round your answer to the nearest whole number.

First, we need to calculate t, we know that t=2022-2018=4 years.

The function is p(t)=3050(1.05)t
p(t=4)=3050(1.05)4=3707.29 rounded 3707

The net increase is 3707-3050=657.
From the year 2018 to the year 2022 the net increase of the population was 657 people.

Another way to solve this example is to calculate the coefficient after 4 years (that is 1.054-1) and then multiply it by the initial amount of the population:
3050(1.05)4-3050=3050(1.054-1)=657.

Compounding interest word problems

The function for compounding interest is p(t)= p₀(1+r)t with the following components:

a- represents the initial amount that is deposited for saving

t- represents the time the money is deposited in the saving account (usually measured in years or months)

r- represents the interest for each time period (decimal fraction)

Consider this example:

A saving account yields 1% interest annually. If David deposits 1000 dollars in this account, how much interest will he earn after 3 years?

The formula for a compounding interest is p(t)=p₀(1+r)t
a=1000
t=1
b=1+0.01=1.01
f(t)=a(b)t=1000*1.013=1030.3 dollars

The interest is equal to the money in the saving account after 3 years minus the initial deposit:
1030.3-1000=30.3 dollars

Note that the answer is not 1%*3*1000=30 because the interest is calculated annually  and the deposit is growing each year (each year we add the interest that was accumulated during the year to the deposit).

After one year the deposit will be 1000*1.01=1010 dollars. The deposit for calculating the interest of the second year will be 1010 dollars and not 1000 dollars.

After 2 years the deposit will be 1010*1.01=1020.1 dollars. The deposit for calculating the interest of the third year will be 1020.1 dollars and not 1000 dollars.

After 3 years the deposit will be 1020.1*1.01=1030.3 dollars, therefore the interest is 30.3 dollars and not 30 dollars.

The rules for changing time units in exponential expressions

How to change the time units in exponential expressions? In order to change the time units, we need to rewrite the exponent without changing the base.

Changing to exponential expressions with bigger time units:
When we need to change the time to bigger units, we need to make the exponent bigger. For example: if the interest for 1 month is 1%, the interest for 1 year is 1*12=12%.

Below are examples for changing the time unit in the exponent of the formula:
The formula for converting months (t) to years is: f(b)=a(b)t*12.
The formula for converting days (t) to years is: f(b)=a(b)t*365.
The formula for converting minutes (t) to hours is: f(b)=a(b)t*60.
The formula for converting hours (t) to days is: f(b)=a(b)t*24.

Changing to exponential expressions with smaller time units:
When we need to change the time to smaller units, we need to make the exponent smaller. For example: if the interest for 1 year is 24%, the interest for 1 month is 24/12=2%.

Below are examples for changing the time unit in the exponent of the formula:
The formula for converting years (t) to month is: f(b)=a(b)t/12.
The formula for converting years (t) to days is: f(b)=a(b)t/365.
The formula for converting hours(t) to minutes is: f(b)=a(b)t/60.
The formula for converting days (t) to hours is: f(b)=a(b)t/24.

Consider this example:

The function s=100*1.2y models the amount of money in a portfolio after y years.

What is the amount of money in the portfolio after 4 month?

We need to rewrite years as months. What is the connection between 1 year and 4 months? 4 months are equal to 4/12=0.33 years, therefore instead of writing y in the exponent we will write there y*0.33.

The function for 4 months period will be s=100*1.2y*0.33.
Since the time unit of 4 months is smaller than the time unit of 1 year, we made the exponent smaller (0.33*y<y).

You just finished studying exponential expressions topic, the fourth topic of passport to advanced mathematics subscore!

Continue studying the next passport to advanced mathematics subscore topic- graphing exponential functions.