Exponents and Radicals: Laws, Rules, Examples and Practice

What is an exponent?

An exponent (or power) is a small raised number that says how many times the base is multiplied by itself: a^n=a· a… a, with n factors of a. In 2^5 the base is 2 and the exponent is 5.

An exponent of 1 changes nothing (a^1=a). The words "squared" and "cubed" come from geometry: 5^2 is the area of a square with side 5, and 10^3 is the volume of a cube with side 10.

Powers grow fast. Each time the exponent goes up by 1, the value is multiplied by the base again: 2^10=1,024, 2^20=1,048,576 and 2^30 is already over a billion. That is why exponents are the language of growth, of computer memory and of very large and very small quantities, and why the successive powers of a number form a geometric sequence.

What are the laws of exponents?

There are five laws of exponents: to multiply powers with the same base add the exponents, to divide subtract them, to raise a power to a power multiply them, and a power of a product or of a quotient is the product or quotient of the powers.

Why the product law works. Count the factors: 2^3·2^4=(2·2·2)(2·2·2·2) has 3+4=7 factors of 2, so it is 2^7. The other laws come from counting in the same way.

The laws need one thing in common. The product and quotient laws combine powers of the same base: 2^3·2^4=2^7, but 2^3·3^4 cannot be combined into one power. Try the laws on numbers of your own below, and see both sides calculated exactly.

What do zero, negative and fractional exponents mean?

a^0=1 for any a≠0, a^-n=(1)/(a^n), and a^1/n=a^(1/n): these are not extra rules to memorize but the only values that keep the laws of exponents true. Counting factors cannot explain an exponent of 0 or of −3, so mathematicians ask what value the laws force.

• Zero. The product law says a^m· a^0=a^m+0=a^m. Multiplying a^m by a^0 changes nothing, so a^0 must be 1.
• Negative. a^n· a^-n=a^0=1. Something that multiplies a^n to give 1 is its reciprocal, so a^-n=(1)/(a^n). A negative exponent does not make the number negative: 2^-3=\dfrac18, not -8.
• Fractional. \left(a^1/n\right)^n=a^\frac1n· n=a^1=a. A number whose n-th power is a is an n-th root, so a^1/n=a^(1/n). More generally a^m/n=\left(a^1/n\right)^m=a^m^(1/n), so any rational number can be an exponent.

The pattern is easiest to see by walking down the powers of 2. Each step down divides by 2:

2^4=16, 2^3=8, 2^2=4, 2^1=2, 2^0=1, 2^-1=\frac12, 2^-2=\frac14, 2^-3=\frac18.

Use the widget to try another base.

What is a radical (root)?

The n-th root of a, written a^(1/n), is the number that gives a when it is raised to the power n; √a (index 2) is the square root and a^(1/3) is the cube root. In a^(1/n) the symbol √( ) is the radical sign, a is the radicand and n is the index.

• Square root: √(49)=7 because 7^2=49.
• Cube root: 125^(1/3)=5 because 5^3=125, and -8^(1/3)=-2 because (-2)^3=-8.
• Principal root: √9=3, not ±3. The radical sign always means the non-negative root. The equation x^2=9 has two solutions, x=3 and x=-3, which is why a solution step writes x=±√9.
• Even and odd roots: an even root of a negative number is not real (√(-4)), but an odd root of a negative number is (-27^(1/3)=-3).
• A root of a square: √(x^2)=|x|, not x, because the result may not be negative: √((-5)^2)=√(25)=5.
• A root in geometry: the diagonal of a unit square is √(1^2+1^2)=√2, by the Pythagorean theorem.

A root of a whole number is either a whole number (when the number is a perfect power, as in the table) or an irrational number: for a positive integer a, √a is irrational exactly when a is not a perfect square. That is why the decimal expansions of √2, √3 and √(72) are infinite and never periodic; the article on irrational numbers shows how to test any root.

How do you simplify a radical?

To simplify a radical, factor the number under the root, take every group of n equal factors out of an n-th root, and multiply what comes out: √(72)=√(2^3·3^2)=3·2√2=6√2. Follow these steps, shown for √(72):

1. Factor the radicand into primes: 72=2·2·2·3·3.
2. Group equal primes in sets as large as the index (pairs for a square root): (2·2)·2·(3·3).
3. Take one prime out for each group; what has no partner stays inside: the pair of 2s gives 2, the pair of 3s gives 3, and one 2 stays under the root.
4. Multiply what came out: 2·3=6, so √(72)=6√2.

The rules behind it, for a,b\ge0: √(ab)=√a·√b and √(a/b)=(√a)/(√b) for b>0. For a cube root take groups of three: 54^(1/3)=2·3^3^(1/3)=32^(1/3).

Adding radicals. Only like radicals, those with the same index and radicand, combine: 3√2+5√2=8√2. Two different-looking roots may turn out alike once simplified: √(50)+√8=5√2+2√2=7√2. But √(a+b) is not √a+√b: √(9+16)=5, not 3+4.

Try any number below. Switch the index to see cube roots and higher roots.

How do you rationalize a denominator?

To rationalize a denominator, multiply the top and bottom of the fraction by a number that removes the root from the bottom: by the root itself for (a)/(√b), and by the conjugate for (a)/(p+√q). Multiplying top and bottom by the same non-zero number does not change the value of a fraction.

• A single root. (6)/(√3)=(6·√3)/(√3·√3)=(6√3)/(3)=2√3, because √3·√3=3.
• A sum or difference. The conjugate of p+√q is p-√q (flip the sign). Their product is a difference of squares, which has no root left: (p+√q)(p-√q)=p^2-q.

For example (1)/(1+√2)=(1·(1-√2))/((1+√2)(1-√2))=(1-√2)/(1-2)=(1-√2)/(-1)=√2-1.

How to rationalize a denominator, in short: (1) choose the root, or the conjugate when the bottom is a sum or a difference; (2) multiply the top and the bottom by it; (3) simplify, and cancel any common factor.

Rationalizing is a convention, not a correction: (1)/(√2) and (√2)/(2) are the same number. The second is the expected form, and it is easier to estimate by hand, since √2≈1.414 gives about 0.707.

How are radicals and fractional exponents connected?

A radical is a fractional exponent: a^(1/n)=a^1/n and a^m^(1/n)=\left(a^(1/n)\right)^m=a^m/n, so every law of exponents also works for roots.

How to evaluate a^m/n:

1. Take the n-th root of the base (the denominator of the exponent): 8^2/3 starts with 8^(1/3)=2.
2. Raise the result to the power m (the numerator): 2^2=4.
3. If the exponent is negative, take the reciprocal: 8^-2/3=(1)/(8^2/3)=\dfrac14.

Taking the root first keeps the numbers small, which is why this order is best by hand.

The exponent form is also the only way to combine roots with different indexes. Since √2=2^1/2 and 2^(1/3)=2^1/3, the product law gives √2·2^(1/3)=2^1/2+1/3=2^5/6=32^(1/6). Try your own below.

How do you solve equations with exponents and roots?

To solve an exponent equation write both sides with the same base and equate the exponents; to solve a radical equation isolate the root, raise both sides to the index, solve, and check every answer, because raising to a power can create false solutions.

Exponent equations (same base).

• 2^x=32 ⇒ 2^x=2^5 ⇒ x=5.
• 9^x=27, so (3^2)^x=3^3, so 3^2x=3^3, so 2x=3, so x=\dfrac32.
• A power on the unknown: x^3=27 ⇒ x=3 (an odd power has one real root), but x^2=9 ⇒ x=±3 (an even power has two).

Radical equations. √(x+1)=3 ⇒ x+1=9 ⇒ x=8, and the check √(8+1)=3 holds.

Always check. Squaring both sides can add answers that do not fit the original equation, called extraneous solutions. Take √(x+2)=x:

1. Square both sides: x+2=x^2.
2. Rearrange: x^2-x-2=0, which factorizes as (x-2)(x+1)=0, so x=2 or x=-1.
3. Check x=2: √4=2 ✓. Check x=-1: √1=1, which is not -1 ✗.

Only x=2 is a solution. The root sign is never negative, so the right side cannot be negative either.

How do you write numbers in scientific notation?

Scientific notation writes a number as a×10^k with 1\le a<10 and k an integer: move the decimal point until one non-zero digit is left of it, and let k count the places moved. Moving left makes k positive and moving right makes it negative.

The first three values are exact by definition in the International System of Units. Scientific notation is the laws of exponents at work. To multiply, multiply the numbers and add the exponents: (3×10^5)(2×10^4)=6×10^9. To divide, divide the numbers and subtract the exponents. To add, first give both numbers the same power of 10.

How do you calculate exponents and roots in Python and JavaScript?

In Python and JavaScript the power operator is a double asterisk, not the caret (which is bitwise XOR), and a root is a power of 1/n or a function such as math.sqrt and Math.cbrt.

>>> 2 ** 10
1024
>>> 2 ** -3
0.125
>>> 16 ** 0.5
4.0
>>> 27 ** (1/3)
3.0
>>> import math
>>> math.sqrt(72)
8.48528137423857
>>> math.isqrt(72)      # whole-number square root, rounded down
8
>>> (-8) ** (1/3)       # a negative base with a fractional exponent
(1.0000000000000002+1.7320508075688772j)
>>> 2 ^ 3               # XOR, not a power
1

2 ** 10               // 1024
2 ** -3               // 0.125
Math.sqrt(72)         // 8.48528137423857
Math.cbrt(-8)         // -2
(-8) ** (1 / 3)       // NaN
(-2) ** 2             // 4   (-2 ** 2 is a SyntaxError)
Math.sqrt(2) ** 2     // 2.0000000000000004
2n ** 100n            // 1267650600228229401496703205376n (BigInt stays exact)

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

int isqrt(int n) {                                   // the whole-number square root, rounded down
  int r = 0;
  while ((r + 1) * (r + 1) <= n) r++;
  return r;
}

int main() {
  printf("%.0f %.3f %.0f\n", pow(2, 10), pow(2, -3), pow(16, 0.5));
  printf("%.12f\n", sqrt(72));
  cout << isqrt(72) << endl;
  printf("%.12f %.12f\n", pow(27, 1.0 / 3), -pow(8, 1.0 / 3));   // a cube root of a negative: take the sign out
  // pow(-8, 1.0 / 3) is nan: a negative base needs a whole-number exponent
  cout << (2 ^ 3) << endl;                             // XOR, not a power
  printf("%.16f\n", sqrt(2) * sqrt(2));                // rounding, not a bug
  int p = 1;
  for (int i = 0; i < 30; i++) p *= 2;                 // 2^30 fits in an int; 2^31 would overflow
  cout << p << endl;
  return 0;
}

/* output:
1024 0.125 4
8.485281374239
8
3.000000000000 -2.000000000000
1
2.0000000000000004
1073741824
*/

The pitfalls, in order of how often they bite:

Use math.isqrt in Python when you need the whole-number part of a square root: it is exact for any size of integer, while math.sqrt works with floating-point numbers (floats).

What are the common mistakes with exponents and radicals?

The most common mistakes with exponents and radicals are the nine below, each with the correct statement and a number that shows why.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: exponents and radicals at a glance

• Exponent: a^n is n factors of a; the base is a and the exponent is n.
• Five laws: a^ma^n=a^m+n, (a^m)/(a^n)=a^m-n, (a^m)^n=a^mn, (ab)^n=a^nb^n, \left(\dfrac ab\right)^n=(a^n)/(b^n).
• Forced values: a^0=1 (a≠0), a^-n=\dfrac1a^n, a^1/n=a^(1/n), a^m/n=\left(a^(1/n)\right)^m.
• Roots: √(x^2)=|x|; an even root of a negative number is not real; √(a+b)≠√a+√b.
• Simplify by taking groups of n equal prime factors out of an n-th root; rationalize with the root or the conjugate.
• Equations: equal bases give equal exponents; check every radical solution.
• Scientific notation: a×10^k with 1\le a<10; in code use **, not ^.

Glossary: key terms

Exponent (power)

The small raised number that says how many times the base is multiplied by itself, as the 5 in 2 to the power 5.

Base

The number that is multiplied by itself in a power, as the 2 in 2 to the power 5.

Radical (root)

The n-th root of a number is the value that gives that number when raised to the power n, written with the radical sign.

Radicand

The number written under the radical sign, such as 72 in the square root of 72.

Index

The small number on the radical sign that says which root is taken; 2 is a square root and 3 is a cube root.

Principal square root

The non-negative square root, which is what the radical sign means: the square root of 9 is 3, not plus or minus 3.

Perfect square

A whole number that is the square of a whole number, such as 1, 4, 9, 16 and 25.

Rational exponent

A fractional exponent m over n, meaning the n-th root of the base raised to the power m.

Conjugate

For p plus the square root of q, the expression p minus the square root of q; their product has no root.

Rationalizing the denominator

Multiplying the top and bottom of a fraction by a number that removes the root from the denominator.

Scientific notation

A way of writing a number as a times 10 to the power k, with a at least 1 and less than 10.

Extraneous solution

A value that solves the equation after squaring both sides but fails in the original equation.

Frequently asked questions

References

1. James Stewart, Lothar Redlin and Saleem Watson (2016). Precalculus: Mathematics for Calculus (7th ed.). Cengage Learning.
2. Walter Rudin (1976). Principles of Mathematical Analysis (3rd ed.), chapter 1 (the existence of n-th roots of positive reals). McGraw-Hill.
3. Florian Cajori (1928). A History of Mathematical Notations. Open Court Publishing Company.
4. René Descartes (1637). La Géométrie.
5. Christoff Rudolff (1525). Behend und hübsch Rechnung durch die kunstreichen regeln Algebre, so gemeincklich die Coss genennt werden.
6. Bureau International des Poids et Mesures (2019). The International System of Units (SI), 9th edition. BIPM. https://www.bipm.org/en/publications/si-brochure.
7. Python Software Foundation. The Python Language Reference: the power operator. docs.python.org. https://docs.python.org/3/reference/expressions.html#the-power-operator.
8. Ecma International. ECMAScript Language Specification: the exponentiation operator. tc39.es. https://tc39.es/ecma262/#sec-exp-operator.

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