Exponents and Radicals

The exponent rules

An exponent counts equal factors: x⁵ = x · x · x · x · x, five factors of x. In x⁵ the base is x and the exponent is 5. Every rule below comes from counting factors. In this table m and n are whole numbers, and no base is 0 when it ends up in a denominator.

Why add in the product rule? x⁴ · x³ is four factors of x followed by three more, seven in all. Why subtract in the quotient rule? In x⁹/x⁴, four factors on the bottom cancel four on the top, leaving five.

Zero and negative exponents

Read this table from the top down. Each time the exponent goes down by 1, the value is divided by 2. Keep going past 2¹ and the pattern tells you what 2⁰ and negative exponents have to mean.

So for any base x that is not 0, x⁰ = 1 and x⁻ⁿ = 1/xⁿ. This course does not give 0⁰ a value. The quotient rule agrees: x³/x³ = x⁰, and anything divided by itself is 1.

Scientific notation

Very large and very small numbers are easier to read and compare as a number times a power of 10. A number is in scientific notation when it is written as a × 10ⁿ, where a is at least 1 and less than 10 and n is an integer. A positive n makes a big number; a negative n makes a small one.

Some real examples. The astronomical unit, the standard distance used for Earth to the Sun, is defined as exactly 149,597,870,700 meters, about 1.496 × 10¹¹ meters. Light travels exactly 299,792,458 meters per second, about 2.998 × 10⁸ meters per second. The charge on one electron, the elementary charge, is exactly 1.602176634 × 10⁻¹⁹ coulombs. Written out, that is about 0.00000000000000000016 coulombs, with 18 zeros after the decimal point before the 16.

How long does sunlight take to reach Earth? Time is distance divided by speed, so divide the astronomical unit by the speed of light, using the rounded values above.

Square roots and cube roots

A square root undoes squaring: √49 = 7 because 7² = 49. The number −7 also squares to 49, but the sign √ always means the principal root, the one that is not negative. A cube root undoes cubing: ∛125 = 5 because 5³ = 125. Cube roots of negative numbers are real numbers, for example ∛−27 = −3, because (−3)³ = −27. But √−9 is not a real number, since no real number squares to a negative.

Knowing these by heart makes simplifying much faster. On this page every variable under a radical is assumed to be 0 or positive, so √x² = x and √x⁶ = x³.

Simplifying radicals

The product property says √ab = √a · √b when a and b are not negative. To simplify a square root, split the number under the sign into a perfect square times what is left, then take the square root of the perfect square. A square root is simplified when the number under it has no perfect-square factor except 1, there is no fraction under it, and there is no radical in a denominator.

The quotient property works the same way for division: √a/b = √a/√b when a is not negative and b is positive. For example, √9/16 = 3/4.

Adding, multiplying and rationalizing

You can add or subtract radicals only when they are like radicals: the same root of the same number, just as 3x + 5x = 8x. Simplify each radical first, because some like radicals are hiding.

To multiply, use the product property the other way round: multiply the numbers outside together and the numbers inside together, then simplify.

To rationalize a denominator, multiply the top and the bottom by the radical in the denominator. That is multiplying by 1, so the value does not change.

Rational exponents

What should 9^1/2 mean? Squaring it should multiply the exponents, and 1/2 · 2 = 1, so (9^1/2)² = 9. That makes 9^1/2 the number whose square is 9. That is √9 = 3. In the same way x^1/3 = ∛x, and x^1/n is the nth root of x.

A fraction exponent m/n does two jobs: the denominator n is the root and the numerator m is the power. Take the root first to keep the numbers small. A negative fraction exponent still means the reciprocal.

Going the other way, a radical can be written as a power: √x³ = x^3/2 and ∛x² = x^2/3.

Common mistakes

Practice problems

Printable worksheet (PDF)

Work each one on paper, then open the answer to check your work. Leave answers with positive exponents and simplified radicals.

1. Simplify: 2x⁵ · 7x³
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   (2 · 7)x^{5 + 3}Multiply the numbers and add the exponents14x⁸
2. Simplify: (3x⁴)³
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   3³ · (x⁴)³Raise each factor to the power 327x¹²3³ = 27 and 4 · 3 = 12
3. Simplify: 24x⁹/8x³
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   (24/8)x^{9 − 3}Divide the numbers and subtract the exponents3x⁶
4. Simplify, with positive exponents: x² · x⁻⁶
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   x^{2 − 6}Add the exponents 2 and −6x⁻⁴1/x⁴Positive exponent
5. Evaluate: (1/2)⁻³
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   2³Flip the fraction and make the exponent positive8
6. Write in scientific notation: 0.000075
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   7.5 × 10⁻⁵Move the decimal point 5 places to the right
7. Multiply, in scientific notation: (6 × 10⁵)(4 × 10⁻²)
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   (6 · 4) × 10^{5 − 2}Multiply the numbers and add the exponents 5 and −224 × 10³24 is not less than 102.4 × 10⁴24 = 2.4 × 10
8. Simplify: √48x⁵
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   √16x⁴ · 3xSplit off the perfect squares 16 and x⁴4x²√3x√16x⁴ = 4x²
9. Simplify: 2√12 + √27
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   2 · 2√3 + 3√3√12 = 2√3 and √27 = 3√34√3 + 3√3Like radicals7√3
10. Rationalize: 10/√5
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    10√5/√5 · √5Multiply the top and the bottom by √510√5/52√510/5 = 2
11. Evaluate: 27^2/3
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    (∛27)²Cube root, then square3²∛27 = 39

History: signs for roots and powers

The symbols on this page took centuries to settle. According to Jeff Miller’s notes on the earliest uses of mathematical symbols, hosted by MacTutor, the radical sign first appeared in 1525, in Die Coss by the German writer Christoff Rudolff. His sign had no bar on top; René Descartes added the bar in La Géométrie in 1637. Writing a small 3 in the opening of the sign for a cube root was suggested by Albert Girard in 1629. A popular story says the sign √ is a stretched letter r, for the Latin radix, “root”. Leonhard Euler held that view, but the historian Florian Cajori argued against it, so it is best treated as unsettled.

Exponents grew up in stages. In the fourteenth century Nicole Oresme used numbers to show powers, and in his Algorismus proportionum he even wrote fractional powers, though not in today’s form. In 1484 Nicolas Chuquet’s Triparty used raised numbers, including negative ones, in a notation of his own. Raised exponents in the modern style, for positive whole numbers, appear in Descartes’s Géométrie of 1637. John Wallis suggested negative exponents and wrote about fractional ones without using them. Isaac Newton wrote both negative and fractional exponents in today’s notation in a June 1676 letter to Henry Oldenburg, secretary of the Royal Society.

Nobody knows who first used scientific notation. Miller’s notes find powers of ten in the work of physicists measuring electricity in the 1860s, before units like the volt and the ohm were standardized.

Connections to polynomials

In the Polynomials and Factoring unit you already used the product rule every time you multiplied monomials: 3x² · 4x⁵ = 12x⁷. Factoring out a greatest common factor is the quotient rule at work, because each term is divided by the factor: 12x⁵ − 8x³ = 4x³(3x² − 2).

The exponents in a polynomial are always whole numbers. That is why x² + 3x⁻¹ and √x + 1 are not polynomials: one has a negative exponent, and the other is x^1/2 + 1. Radicals also finish problems from the Quadratic Equations unit. Solving x² = 50 gives x = √50 or x = −√50, and simplifying gives x = 5√2 or x = −5√2.

Sources

• Jeff Miller, “Earliest Uses of Symbols of Operation”, hosted by MacTutor History of Mathematics, University of St Andrews: Symbols of operation (the radical sign in Rudolff’s Die Coss, 1525; Descartes’s bar, 1637; Girard’s index, 1629; Euler’s and Cajori’s views on its origin; exponents in Oresme, Chuquet 1484, Descartes 1637, Wallis and Newton’s 1676 letter; early scientific notation).
• J. J. O’Connor and E. F. Robertson, MacTutor: Christoff Rudolff and Nicole Oresme.
• NASA Jet Propulsion Laboratory, Solar System Dynamics: Astrodynamic Parameters (astronomical unit = 149,597,870,700 m, exact).
• NIST Reference on Constants, Units and Uncertainty: speed of light in vacuum (299,792,458 m/s, exact) and elementary charge (1.602176634 × 10⁻¹⁹ C, exact).
• Common Core State Standards for Mathematics: Expressions and Equations, grade 8 (8.EE.A.1 to 8.EE.A.4, exponent rules, roots and scientific notation) and The Real Number System (N-RN.A.1 and N-RN.A.2, rational exponents), which this lesson supports.

Credits: all diagrams on this page are original SVG drawings made for Jugami. Every rule, example and step was checked by a script.