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Browse High School Exponents
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Negative exponents.
    Scientific notation.

Cube Root by Hand [01/23/1998]
How do you calculate the cube root of a number without using a calculator?

Dividing Negative Exponents [10/14/1998]
Why does a negative exponent become positive when you divide by it?

Do Irrational Roots Always Occur in Conjugate Pairs? [03/13/2002]
A colleague and I have been puzzled by the equation x^3+4x^2+3x+1=0. Upon graphing, there appear to be two imaginary solutions and one real solution...

Exponents and Negative numbers [03/02/1997]
Why wouldn't the answer to -3 squared, standing alone and without parenthesis or brackets, be -9?

Heron's Method for Finding Square Roots by Hand [12/18/2001]
How do you get the square root of a number manually without a table or calculator?

Negative Exponents Explained [12/29/2001]
I have a problem with exponents like 5 to the negative second.

Negative Square Roots on Calculators [10/30/1995]
Every time I try to calculate on my calculator the square root of a negative number (-4, for example) I get the message "Error". Why?

Palindromic Squares [07/28/1997]
Do you know any numbers besides 14641 where both the number and its square root read the same left to right as right to left?

Proof that Sqrt(2) is Irrational [8/29/1996]
What is the proof that the square root of 2 is irrational? I recall it begins with the assumption that sqrt(2) is rational...

Square Root of a Negative Number [01/25/1997]
Is it possible to find the square root of a negative number and, if so, to what number system do these square roots belong?

Why Simplify Square Roots? [05/17/1999]
We have been trained to simplify square roots, but we are now looking at the reasons why we should continue to teach the process.

1005! / 10^n: A Number Puzzle [9/11/1996]
Find the largest positive integer, n, such that 1005! is divisible by 10 to the power of n.

21^100 - Last Two Digits [09/04/1997]
What are the last two digits of 21 to the 100th power?

2/3 Power using a Calculator [07/05/2001]
Can -512 be the answer to the equation x^(2/3) = 64?

24^25 or 25^24: Proving Which Is Larger [03/12/2015]
An exponent and its base differ by one. Given the two arrangements of the smaller raised to the larger and vice versa, an adult wonders how to compare their relative sizes without actually evaluating them. Doctor Ali applies some offsetting exponentiation en route to a proof via calculus.

2^99 Divided by 99 [08/27/2001]
What is the remainder when you divide 2^99 by 99?

Adding Square Roots [10/31/1995]
I don't understand how to add 3 Radical 81 + 2 Radical 25.

Algebra 2: Exponents [01/07/1998]
I need to find out what "x" is in this problem: x^x^3 = 3.

Algorithm for Simple Fractional Exponents [03/14/2002]
How can I explain to my program how to resolve simple fractional exponents using only addition, subtraction, multiplication, division, square, and square root?

Alternative Formulas for Growth and Decay [08/02/2002]
I have two formulas for growth and decay: q = q_0*e^(rt), and q = q_0*a^t. What is the difference between them?

Applications of Non-antidifferentiable Functions [01/29/1999]
What are some of the real life applications of the functions sin (x)/ x, e^(-x^2), and sin (x^2)?

Approaching Zero and Losing the Plot [11/11/2010]
Looking near the origin at plots of y = x^n for ever tinier n, a student wonders why y = x^0 does not equal zero. By emphasizing two different limits, Doctor Ali gets the student back into line -- specifically, y = 1.

Are Exponents Associative? [02/05/2002]
How much is 2^(i) or x^(i) ?

Are Negative Exponents Like Other Exponents? [08/21/2002]
Is there a general rule for doing all exponents, or does a negative exponent have nothing in common with positive exponents?

Arithmetic vs. Exponential Increases [05/06/1999]
What does "....the work produced... will increase exponentially rather than arithmetically" mean?

Base e, In, Log [03/08/2002]
Problems in base e. For example: 3e^(2x-1) = 7, or e^(x+1) = 8.

Base of an Exponential Function [09/15/2001]
Why can't the base of an exponential function be negative?

Binary Exponentiation [09/13/2001]
Show how to compute 5^21 using 6 multiplications. How many multiplications are needed to compute 7^7024?

Calculate the Derivative for Sqrt(x^5) [03/09/2003]
Why does the calculator fail when I calculate the derivative for sqrt(x^5)?

Coal Consumption [01/13/1997]
How long will coal reserves last if consumption increase at the rate of 3.1 percent per year?

Code with Only a String of 1's [10/28/2002]
You need to send a numeric code to someone, such as one used to unlock a padlock.

Combining Positive and Negative Exponents [06/30/1999]
When you simplify exponential expressions using the power rule, how do you combine positive and negative exponents in the numerator and denominator?

Comparing n^k and k^n [04/03/2007]
If n > k, which is greater, n^k or k^n? Is there any general rule to fit this situation?

Comparing Numbers With Huge Exponents [05/23/2002]
Which is larger, 1999^1999 or 2000^1998?

Complex Exponentiation [02/20/2015]
A teen struggles to evaluate expressions with complex numbers in their exponents. Using De Moivre's Theorem and polar coordinates, Doctors Ali and Peterson break down the method.

Complex Radicals [11/27/2001]
I understand the square root, but how do you use the cubed root?

Conjugates with Square Roots [1/29/1996]
I don't understand what conjugates are in relation to square roots, and what good they do.

Cube Roots of Numbers [11/05/1997]
If you take i (sqrt(-1)), the cube root is -i, but since x^3 = i is degree three there should be three different values of x. What are they?

Cubic Equations [01/19/1998]
How would you solve a problem like 3x^3+x^2+15x+27 = 0?

Cubic Equations in One Formula [11/27/2001]
I know the formula for solving quadratic equations of the type ax^2+bx+c=0, but is there a formula for cubic equations?

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