exponent - The power to raise a number

(2n)^3 without exponents

(2n)^3 without exponents
This can be written as follows:
2^3 * n^3
[B]8n^3[/B]

(n^2)^3 without exponents

(n^2)^3 without exponents
This expression evaluates to:
n^(2 *3)
n^6
To write this without exponents, we expand n times itself 6 times:
[B]n * n * n * n * n * n[/B]

11 to the power of 6 multiply 11 to the power of 3

11 to the power of 6 multiply 11 to the power of 3
Take this in parts.
[U]Step 1: 11 to the power of 6 means we raise 11 to the 6th power using exponents:[/U]
11^6
[U]Step 2: 11 to the power of 3 means we raise 11 to the 3rd power using exponents:[/U]
11^3
[U]Step 3: Multiply each term together:[/U]
11^6 * 11^3
[U]Step 4: Simplify[/U]
Because we have 2 numbers that are the same, in this case, 11, we can add the exponents together when multiplying:
11^(6 + 3)
[B]11^9
[MEDIA=youtube]gCxVq7LqyHk[/MEDIA][/B]

2^n = 4^(n - 3)

2^n = 4^(n - 3)
2^n = (2^2)^(n - 3)
(2^2)^(n - 3) = 2^2(n - 3)
2^n= 2^2(n - 3)
Comparing exponents, we see that:
n = 2(n - 3)
n = 2n - 6
Subtract n from each side:
n - n = 2n - n - 6
0 = n - 6
n = [B]6[/B]

3 to the power of 2 times 3 to the power of x equals 3 to the power of 7

3 to the power of 2 times 3 to the power of x equals 3 to the power of 7.
Write this out:
3^2 * 3^x = 3^7
When we multiply matching coefficients, we add exponents, so we have:
3^(2 + x) = 3^7
Therefore, 2 + x = 7. To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=2%2Bx%3D7&pl=Solve']type it into our search engine[/URL] and we get:
x = [B]5[/B]

8 to the power of x over 2 to the power of y

8 to the power of x over 2 to the power of y
Step 1: 8 to the power of x means we take 8 and raise it to an exponent of x:
8^x
Step 2: 2 to the power of y means we take 2 and raise it to an exponent of y:
2^y
Step 3: The word [I]over[/I] means a quotient, also known as divided by, so we have:
[B]8^x/2^y
[MEDIA=youtube]SPQKOt5EoqA[/MEDIA][/B]

A company has 3,100 employees and is expected to grow at a rate of 0.04 for the next six years. How

A company has 3,100 employees and is expected to grow at a rate of 0.04 for the next six years. How many employees will they have in 6 years? Round to the nearest whole number.
We build the following exponential equation:
Final Balance = Initial Balance * (1 + growth rate)^time
Final Balance = 3100(1.04)^6
Final Balance = 3100 * 1.2653190185
Final Balance = 3922.48895734
The problem asks us to round to the nearest whole number. Since 0.488 is less than 0.5, we round [U]down.[/U]
Final Balance = [B]3,922[/B]

A super deadly strain of bacteria is causing the zombie population to double every day. Currently, t

A super deadly strain of bacteria is causing the zombie population to double every day. Currently, there are 25 zombies. After how many days will there be over 25,000 zombies?
We set up our exponential function where n is the number of days after today:
Z(n) = 25 * 2^n
We want to know n where Z(n) = 25,000.
25 * 2^n = 25,000
Divide each side of the equation by 25, to isolate 2^n:
25 * 2^n / 25 = 25,000 / 25
The 25's cancel on the left side, so we have:
2^n = 1,000
Take the natural log of each side to isolate n:
Ln(2^n) = Ln(1000)
There exists a logarithmic identity which states: Ln(a^n) = n * Ln(a). In this case, a = 2, so we have:
n * Ln(2) = Ln(1,000)
0.69315n = 6.9077
[URL='https://www.mathcelebrity.com/1unk.php?num=0.69315n%3D6.9077&pl=Solve']Type this equation into our search engine[/URL], we get:
[B]n = 9.9657 days ~ 10 days[/B]

A virus is spreading exponentially. The initial amount of people infected is 40 and is increasing at

A virus is spreading exponentially. The initial amount of people infected is 40 and is increasing at a rate of 5% per day. How many people will be infected with the virus after 12 days?
We have an exponential growth equation below V(d) where d is the amount of days, g is the growth percentage, and V(0) is the initial infected people:
V(d) = V(0) * (1 + g/100)^d
Plugging in our numbers, we get:
V(12) = 40 * (1 + 5/100)^12
V(12) = 40 * 1.05^12
V(12) = 40 * 1.79585632602
V(12) = 71.8342530409 or [B]71[/B]

a ^5 x a ^2 without exponents

a ^5 x a ^2 without exponents
When we multiply the same variable or number, we add exponents, so we have:
a^(5 + 2)
a^7
To write a variable raised to an exponent without exponents, we break it up. The formula to do this is:
a^n = a times itself n times
a^7 = [B]a * a * a * a * a * a * a[/B]

add 7 and 2, raise the result to the 6th power, then add what you have to s

add 7 and 2, raise the result to the 6th power, then add what you have to s
Add 7 and 2:
7 + 2
Simplify this, we get:9
Raise the result to the 6th power:
9^6
[URL='https://www.mathcelebrity.com/powersq.php?sqconst=+6&num=9%5E6&pl=Calculate']Simplifying this using our exponent calculator[/URL], we get:
531,441
Now, we add what we have (our result) to s to get our final algebraic expression:
[B]s + 531,441[/B]

add d to 5, raise the result to the 9th power, then subtract what you have from 2

add d to 5, raise the result to the 9th power, then subtract what you have from 2
Add d to 5:
d + 5
Raise the result to the 9th power means we raise (d + 5) to the 9th power using an exponent:
(d + 5)^9
the subtract what we have (the result) from 2:
[B]2 - (d + 5)^9[/B]

Approximate Square Root Using Exponential Identity

Calculates the square root of a positive integer using the Exponential Identity Method

Covariance and Correlation coefficient (r) and Least Squares Method and Exponential Fit

Given two distributions X and Y, this calculates the following:

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r^{2}

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

* Covariance of X and Y denoted Cov(X,Y)

* The correlation coefficient r.

* Using the least squares method, this shows the least squares regression line (Linear Fit) and Confidence Intervals of α and Β (90% - 99%)

Exponential Fit

* Coefficient of Determination r squared r

* Spearmans rank correlation coefficient

* Wilcoxon Signed Rank test

Customers arrive at the claims counter at the rate of 20 per hour (Poisson distributed). What is th

Customers arrive at the claims counter at the rate of 20 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes?
Use the [I]exponential distribution[/I]
20 per 60 minutes is 1 every 3 minutes
1/λ = 3 so λ = 0.333333333
Using the [URL='http://www.mathcelebrity.com/expodist.php?x=+5&l=0.333333333&pl=CDF']exponential distribution calculator[/URL], we get F(5,0.333333333) = [B]0.811124396848[/B]

divide 8 by t, raise the result to the 7th power

divide 8 by t, raise the result to the 7th power.
We take this algebraic expression in two parts:
1. Divide 8 by t
8/t
2. Raise the result to the 7th power. (This means we use an exponent of 7)
[B](8/t)^7[/B]

double v, raise the result to the 6th power, then multiply what you have by w

double v, raise the result to the 6th power, then multiply what you have by w
Double v means multiply v by 2:
2v
Raise the result to the 6th power, means we use an exponent of 6 on 2v:
(2v)^6
Then multiply what you have by w, means take the result above, and multiply by w:
[B]w(2v)^6[/B]

Equation and Inequalities

Solves an equation or inequality with 1 unknown variable and no exponents as well as certain absolute value equations and inequalities such as |x|=c and |ax| = c where a and c are constants. Solves square root, cube root, and other root equations in the form ax^2=c, ax^2 + b = c. Also solves radical equations in the form asqrt(bx) = c. Also solves open sentences and it will solve one step problems and two step equations. 2 step equations and one step equations and multi step equations

Exponential Distribution

Calculates the Probability Density Function (PDF) and Cumulative Density Function (CDF) of the exponential distribution as well as the mean, variance, standard deviation, and entropy.

Exponential Growth

This solves for any 1 of the 4 items in the exponential growth equation or exponential decay equation, Initial Value (P), Ending Value (A), Rate (r), and Time (t).

Exponential Smoothing

Performs exponential smoothing on a set of data.

Function

Takes various functions (exponential, logarithmic, signum (sign), polynomial, linear with constant of proportionality, constant, absolute value), and classifies them, builds ordered pairs, and finds the y-intercept and x-intercept and domain and range if they exist.

How many times bigger is 3^9 than 3^3

How many times bigger is 3^9 than 3^3
Using exponent rules, we see that:
3^9 = 3^3 * 3^6
So our answer is [B]3^6 times bigger[/B]

If 200 bacteria triple every 1/2 hour, how much bacteria in 3 hours

If 200 bacteria triple every 1/2 hour, how much bacteria in 3 hours
Set up the exponential function B(t) where t is the number of tripling times:
B(d) = 200 * (3^t)
3 hours = 6 (1/2 hour) periods, so we have 6 tripling times. We want to know B(6):
B(6) = 200 * (3^6)
B(6) = 200 * 729
B(6) = [B]145,800[/B]

If 3x - y = 12, what is the value of 8^x/2^y

If 3x - y = 12, what is the value of 8^x/2^y
We know 8 = 2^3
So using a rule of exponents, we have:
(2^3)^x/2^y
2^(3x)/2^y
Using another rule of exponents, we rewrite this fraction as:
2^(3x -y)
We're given 3x - y = 12, so we have:
[B]2^12[/B]

If there are 10^30 grains of sand on Beach A, how many grains of sand are there on a beach the has 1

If there are 10^30 grains of sand on Beach A, how many grains of sand are there on a beach the has 10 times the sand as Beach A? (Express your answer using exponents.)
10^30 * 10 = 10^(30 + 1) = [B]10^31[/B]

index form of (5^3)^6

Index form of (5^3)^6
Index form is written as a number raised to a power.
Let's simplify by multiply the exponents. Since 6*3 = 18, We have:
[B]5^18[/B]

Logarithms

Using the formula Log a_{b} = e, this calculates the 3 pieces of a logarithm equation:

1) Base (b)

2) Exponent

3) Log Result

In addition, it converts

* Expand logarithmic expressions

1) Base (b)

2) Exponent

3) Log Result

In addition, it converts

* Expand logarithmic expressions

Logarithms and Natural Logarithms and Eulers Constant (e)

This calculator does the following:

* Takes the Natural Log base e of a number x Ln(x) → log_{e}x

* Raises e to a power of y, e^{y}

* Performs the change of base rule on log_{b}(x)

* Solves equations in the form b^{cx} = d where b, c, and d are constants and x is any variable a-z

* Solves equations in the form ce^{dx}=b where b, c, and d are constants, e is Eulers Constant = 2.71828182846, and x is any variable a-z

* Exponential form to logarithmic form for expressions such as 5^{3} = 125 to logarithmic form

* Logarithmic form to exponential form for expressions such as Log_{5}125 = 3

* Takes the Natural Log base e of a number x Ln(x) → log

* Raises e to a power of y, e

* Performs the change of base rule on log

* Solves equations in the form b

* Solves equations in the form ce

* Exponential form to logarithmic form for expressions such as 5

* Logarithmic form to exponential form for expressions such as Log

Modular Exponentiation and Successive Squaring

Solves x^{n} mod p using the following methods:

* Modular Exponentiation

* Successive Squaring

* Modular Exponentiation

* Successive Squaring

Oliver invests $1,000 at a fixed rate of 7% compounded monthly, when will his account reach $10,000?

Oliver invests $1,000 at a fixed rate of 7% compounded monthly, when will his account reach $10,000?
7% monthly is:
0.07/12 = .00583
So we have:
1000(1 + .00583)^m = 10000
divide each side by 1000;
(1.00583)^m = 10
Take the natural log of both sides;
LN (1.00583)^m = LN(10)
Use the identity for natural logs and exponents:
m * LN (1.00583) = 2.30258509299
0.00252458479m = 2.30258509299
m = 912.064867899
Round up to [B]913 months[/B]

On January 1st a town has 75,000 people and is growing exponentially by 3% every year. How many peop

On January 1st a town has 75,000 people and is growing exponentially by 3% every year. How many people will live there at the end of 10 years?
[URL='https://www.mathcelebrity.com/population-growth-calculator.php?num=atownhasapopulationof75000andgrowsat3%everyyear.whatwillbethepopulationafter10years&pl=Calculate']Using our population growth calculator[/URL], we get:
[B]100,794[/B]

Polynomial

This calculator will take an expression without division signs and combine like terms.

It will also analyze an polynomial that you enter to identify constant, variables, and exponents. It determines the degree as well.

It will also analyze an polynomial that you enter to identify constant, variables, and exponents. It determines the degree as well.

Population Growth

Determines population growth based on an exponential growth model.

Raise 9 to the 3rd power, subtract d from the result, then divide what you have by c

Raise 9 to the 3rd power, subtract d from the result, then divide what you have by c.
This is an algebraic expression, let's take in parts (or chunks).
Raise 9 to the 3rd power. This means we take 9, and raise it to an exponent of 3
9^3
Subtract d from the result, means we subtract d from 9^3
9^3 - d
Now we divide 9^3 - d by c
[B](9^3 - d) / c[/B]

raise f to the 3rd power, then find the quotient of the result and g

raise f to the 3rd power, then find the quotient of the result and g
Take this algebraic expression in two parts:
[LIST=1]
[*]Raise f to the 3rd power means we take f, and write it with an exponent of 3: f^3
[*]Find the quotient of the result and g. We take f^3, and divide it by g
[/LIST]
[B]f^3/g[/B]

Raise f to the 8th power, divide the result by 5, then multiply 10

Raise f to the 8th power, divide the result by 5, then multiply 10
f to the 8th power means we raise f to the power of 8 using an exponent:
f^8
Divide f^8 by 5
(f^8)/5
Now multiply this by 10:
10(f^8)/5
We can simplify this algebraic expression by dividing 10/5 to get 2 on top:
2[B](f^8)[/B]

raise r to the 8th power then find the product of the result and 3

raise r to the 8th power then find the product of the result and 3
Raise r to the 8th power means we raise r with an exponent of 8:
r^8
The product of the result and 3 means we muliply r^8 by 3
[B]3r^8[/B]

raise t to the 10th power, then find the quotient of the result and s

raise t to the 10th power, then find the quotient of the result and s
Raise t to the 10th power means we use t as our variable and 10 as our exponent:
t^10
The quotient means a fraction, where the numerator is t^10 and the denominator is s:
[B]t^10/s[/B]

raise v to the 9th power, then dividethe result by u

V to the 9th power means we use an exponent:
v^9
Divide that result by u
[B]v^9/u[/B]

raise y to the 10th power, then find the quotient of the result and 2

y to the 10th power means we give y an exponent of 10
y^10
The quotient of y^10 and 2 is:
y^10
-----
2

Rational Exponents - Fractional Indices

This calculator evaluates and simplifies a rational exponent expression in the form a^{b/c} where a is any integer *or* any variable [a-z] while b and c are integers. Also evaluates the product of rational exponents

rewrite without an exponent :4^-2

rewrite without an exponent :4^-2
Since the exponent is negative, we have:
4^-2 = 1 / 4^2
4^-2 = [B]1 / 16[/B]

Simplest Exponent Form

This expresses repeating algebraic expressions such as 3*a*a*a*b*b into simplest exponent form.

Square Roots and Exponents

Given a number (n), or a fraction (n/m), and/or an exponent (x), or product of up to 5 radicals, this determines the following:

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x^{th} power denoted as n^{x} (Write without exponents)

* n raised to the x^{th} power raised to the yth power denoted as (n^{x})^{y} (Write without exponents)

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

* The square root of n denoted as √n

* The square root of the fraction n/m denoted as √n/m

* n raised to the x

* n raised to the x

* Product of up to 5 square roots: √a√b√c√d√e

* Write a numeric expression such as 8x8x8x8x8 in exponential form

The flu is starting to hit Lanberry. Currently, there are 894 people infected, and that number is gr

The flu is starting to hit Lanberry. Currently, there are 894 people infected, and that number is growing at a rate of 5% per day. Overall, how many people will have gotten the flu in 5 days?
Our exponential equation for the Flu at day (d) is:
F(d) = Initial Flu cases * (1 + growth rate)^d
Plugging in d = 5, growth rate of 5% or 0.05, and initial flu cases of 894 we have:
F(5) = 894 * (1 + 0.05)^5
F(5) = 894 * (1.05)^5
F(5) = 894 * 1.2762815625
F(5) = [B]1141[/B]

the university of california tuition in 1990 was $951 and tuition has been increasing by a rate of 2

the university of california tuition in 1990 was $951 and tuition has been increasing by a rate of 26% each year, what is the exponential formula
Let y be the number of years since 1990. We have the formula T(y):
[B]T(y) = 951 * 1.26^y[/B]

What is an Exponent

This lesson walks you through what an exponent is, the product rule for exponents, the quotient rule for exponents, the 0 power rule, the power of a power rule for exponents

What is the annual nominal rate compounded daily for a bond that has an annual yield of 5.4%? Round

What is the annual nominal rate compounded daily for a bond that has an annual yield of 5.4%? Round to three decimal places. Use a 365 day year.
[U]Set up the accumulation equation:[/U]
(1+i)^365 = 1.054
[U]Take the natural log of each side[/U]
365 * Ln(1 + i) = 1.054
Ln(1 + i) = 0.000144089
[U]Use each side as a exponent to eulers constant e[/U]
(1 + i) = e^0.000144089
1 + i = 1.000144099
i = 0.000144099 or [B].0144099%[/B]

When finding the power of a power, you _____________________ the exponents

When finding the power of a power, you _____________________ the exponents
[B]Multiply
[/B]
Example:
(a^b)^c = a^bc

Zombies are doubling every 2 days. If two people are turned into zombies today, how long will it tak

Zombies are doubling every 2 days. If two people are turned into zombies today, how long will it take for the population of about 600,000 to turn into zombies?
Let d = every 2 days. We set up the exponential equation
2 * 2^d = 600,000
Divide each side by 2:
2^d = 300000
To solve this equation for d, we [URL='https://www.mathcelebrity.com/natlog.php?num=2%5Ed%3D300000&pl=Calculate']type it in our math engine[/URL] and we get
d = 18.19 (2 day periods)
18.19 * days per period = 36.38 total days
Most problems like this ask you to round to full days, so we round up to [B]37 days[/B].