A radical is an expression that uses the root symbol √, such as √25 or 3√x. The number under the symbol is the radicand, and the small number in the notch is the index, which tells which root to take. With no index, it is a square root: √25 = 5.
A radical has three parts: the index (which root), the radical sign, and the radicand (the number inside).
Formal definition and notation
For a positive integern ≥ 2, the radical n√a denotes the principalnth root of a: the real numberb with bn = a, taking b ≥ 0 when n is even. For even n it is defined only for a ≥ 0; for odd n it is defined for all real a. In exponent form, n√a = a1n and n√am = amn.
An exponent repeats multiplication: 43 = 64. A radical undoes it: 3√64 = 4.
Radicalvs.Radicand
The radicand is only the number or expression inside the symbol; in √2x, it is 2x. The radical is the whole expression.
Quick check: 3 questions
Question 1 of 3
In 5√32, what is the index and what is the value?
Show answer and explanation
Answer: B. The small 5 is the index, and 25 = 32, so the value is 2. 32 is the radicand, 6.4 divided 32 by 5, and the last choice mixes up the parts.
Question 2 of 3
Simplify √200.
Show answer and explanation
Answer: D.200 = 100 · 2, so √200 = 10√2. 2√50 is equal but not simplest, 100√2 forgot to take the root of 100, and 20√10 equals √4000.
Question 3 of 3
Which radical is NOT a real number?
Show answer and explanation
Answer: A. No real number squared is −16, so √−16 is not real (it is 4i). 3√−64 = −4 and 5√−1 = −1 are odd roots, and √0 = 0.
Radical: frequently asked questions
What is the difference between a radical and a square root?
A square root is one kind of radical, the one with index 2, usually written without the index: √49 = 7. Radical is the general name for any root written with the √ symbol, including cube roots 3√8 = 2 and fourth roots 4√81 = 3. It also names whole expressions such as 3√x + 1.
How do you simplify a radical?
Factor the radicand and pull out the largest perfect power that matches the index. For square roots, find the largest perfect square factor: √75 = √25 · 3 = 5√3. For cube roots, find perfect cubes: 3√54 = 3√27 · 2 = 33√2. Stop when no such factor remains.
How do radicals relate to exponents?
A radical is a fractional exponent. n√a = a1n, and n√am = amn. So √x = x12 and 3√x2 = x23. Writing radicals as exponents lets you use the exponent rules: √x · 3√x = x12 + 13 = x56, because exponents add when you multiply powers of the same base.
Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1PrincipalThe principal is the original amount of money that is borrowed, deposited or invested, before any interest is added. In the simple interest formula I = Prt, the letter P stands for the principal. Deposit 2,000 dollars at 3% simple interest for 4 years: the principal is 2,000 dollars, the interest is 240 dollars, and the total is 2,240 dollars.Problem Solving · Grade 7, Algebra 1Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. 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Distances, absolute values and principal square roots are always nonnegative.Problem Solving · Grade 6, Grade 7, Algebra 1RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2Negative NumberA negative number is a number less than zero, written with a minus sign, such as −3, −0.5 or −23. On a number line, negative numbers sit to the left of 0. They describe amounts below a starting point: temperatures below zero, depths below sea level, or money owed. Zero itself is neither negative nor positive.Number System · Grade 6, Grade 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7Irrational NumberAn irrational number cannot be written as a fraction of two integers. 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The hypotenuse, c, is the longest side across from the right angle. Legs of 3 and 4 give 9 + 16 = 25, so the hypotenuse is 5.Geometry · Grade 8, Algebra 1