Powers, roots and the real number line

Read the exponent as a count of equal factors

The power 2³ means 2 × 2 × 2 = 8. The base is 2, and the exponent is 3: it counts three factors, each equal to the base. It does not mean 2 × 3, which is 6. Read 2³ as “two cubed” or “two to the third power”. The expression 2⁴ has four factors of 2 and equals 16.

For a positive integer exponent n, aⁿ is a product of n equal factors a. The letters stand for numbers, with n restricted here to 1, 2, 3, and so on. In particular, a¹ = a: one factor is simply the base itself. The same idea applies to zero or a negative number as the base. For example, 0³ = 0 × 0 × 0 = 0. Zero and negative exponents require a further explanation; they are not literal counts of repeated factors.

Connect squares and cubes with area and volume

A square with side length 3 cm has area 3 cm × 3 cm = 9 cm². The square on the unit matters: cm² measures area, while cm measures length. A cube with edge length 3 cm has volume 3 cm × 3 cm × 3 cm = 27 cm³. This is why the second and third powers are called a square and a cube.

For an ideal square of side length s, its area is s²; for an ideal cube of edge length s, its volume is s³. Keep the units with the calculation. Doubling a square’s side from 3 cm to 6 cm changes the area from 9 cm² to 36 cm²: the area becomes four times as large. Doubling a cube’s edge changes the volume from 27 cm³ to 216 cm³: the volume becomes eight times as large. The numerical calculations are exact for the stated lengths; real measurements may carry uncertainty.

Include the sign in the base when you intend to

With (−2)², both factors are −2, so (−2) × (−2) = 4. With (−2)³, one more factor of −2 gives −8. For a negative base and a positive integer exponent, an even number of negative factors can all be paired, producing a positive result. An odd number leaves one negative factor and produces a negative result.

Compare (−2)⁴ = 16 with −2⁴ = −16. Parentheses make −2 the base in the first expression. In the second, the base is 2 and the leading minus takes the opposite of 2⁴. Before calculating, identify the base and the part of the expression to which the exponent applies. A sign on the base and a sign on the exponent have different meanings.

Ask which nonnegative number has the required square

A square tile has area 9 cm². Its side length is 3 cm, because 3 × 3 = 9. We write √9 = 3. The radical symbol √ denotes the principal square root: for a nonnegative input, it selects the nonnegative number whose square equals that input. The number under the radical is the radicand. Thus √0 = 0, √1 = 1 and √16 = 4.

The question x² = 9 is different: find every real number x whose square is 9. Both 3 and −3 work, so its solutions are x = 3 or x = −3, written briefly x = ±3. But √9 is one value, 3, rather than ±3. Squaring loses the original sign: for any real x, √(x²) = |x|. Here |x| is the absolute value, or distance from zero. For example, √((−3)²) = √9 = 3 = |−3|.

No real number has a negative square: a positive or negative number squared is positive, and zero squared is zero. Consequently √(−9) has no real value. Keep the parentheses clear: −√9 = −3 is the opposite of a defined square root, while √(−9) asks for a square root of a negative number. Complex numbers provide a further extension in a later branch.

A cube root can keep a negative sign

The cube root ∛8 is 2 because 2³ = 8. A cube with volume 8 cm³ therefore has edge length 2 cm. The symbol ∛ indicates a cube root; unlike a square root, it asks for a number whose third power gives the input.

A negative input has a real cube root. Since (−2) × (−2) × (−2) = −8, we have ∛(−8) = −2. Each real input has one real cube root, and taking the cube root of x³ returns x, including its sign: ∛(x³) = x. A negative cube root is a signed number, not a negative physical edge length; the geometric example uses positive lengths.

Extend powers by dividing by the same nonzero base

For a nonzero base, decreasing the exponent by one corresponds to dividing by that base. With base 2, the sequence 2³ = 8, 2² = 4, 2¹ = 2 reaches 2⁰ = 1 after one more division by 2. The same reasoning gives a⁰ = 1 whenever a ≠ 0: dividing a¹ by a gives a ÷ a = 1. The symbol ≠ means “is not equal to”. This reasoning cannot be used with base zero because it would divide by zero. We leave 0⁰ outside the rule used in this lesson.

To continue below exponent zero, we need a number smaller than one in this example. The fraction notation 1/8 means 1 ÷ 8: one whole shared into eight equal parts gives one eighth. A slash or a fraction bar represents the division, and the number below the bar cannot be zero. Dividing 1 by 2 three times gives 1/8, so 2⁻³ = 1 ÷ 2³ = 1/8. Check that (1/8) × 8 = 1. We will develop fraction arithmetic in the next lesson.

For any nonzero base a and positive integer n, a⁻ⁿ = 1 ÷ aⁿ. This is the reciprocal of aⁿ: their product is one. A negative exponent does not automatically make a value negative: 2⁻³ is positive, while (−2)⁻³ = 1 ÷ (−8) = −1/8. Zero raised to a negative integer power is undefined in this arithmetic, because its reciprocal would require division by zero.

Combine powers by counting their factors

In 3² × 3³, the first power contributes two factors of 3 and the second contributes three. Altogether there are five: 3² × 3³ = (3 × 3) × (3 × 3 × 3) = 3⁵ = 243. For a real base a and positive integer exponents m and n, aᵐ × aⁿ = aᵐ⁺ⁿ. Add the exponents because you are joining two groups of equal factors.

Keep the conditions and the operation visible. This rule uses the same base and multiplication; it does not say to add exponents in 3² + 3³ or combine different bases in that way. With a nonzero base, the rule also extends to zero and negative integer exponents using one and reciprocals. For example, 2³ × 2⁻³ = 8 × (1/8) = 1 = 2⁰. More rules, fractional exponents, exponential functions and logarithms belong to later lessons.

Change a power, then locate a square root

In power mode, choose an integer base from −4 to 4 and an exponent from 0 to 5. Before looking at the result, predict its sign and count the factors. Compare exponents 2 and 3 with a negative base. Then try exponent zero with a nonzero base and explain why it gives one. The choice 0⁰ is marked as outside this lesson’s rule, rather than being calculated with it.

The decimal labels also use place value. A decimal point separates the whole-number part from smaller units: 1.4 means one whole and four tenths, or 1 + 4/10; 1.41 means one whole and forty-one hundredths, or 1 + 41/100. Each position to the right represents a unit ten times smaller than the one before it. We use these labels to locate roots here and will develop decimal calculations in later lessons.

In square-root mode, choose a whole-number radicand from 0 to 25. Try a perfect square, such as 9, and then a non-square, such as 5. Compare the squares of the displayed endpoints to the radicand. Increase the refinement from whole units to tenths, hundredths or thousandths. The bounds become more precise, while the exact root stays the same. A finite decimal value and a drawn marker for an irrational root are approximations; they do not turn it into an exact finite decimal.

Use exact squares to bound an unknown root

Since 1² = 1 and 2² = 4, and 1 < 2 < 4, the nonnegative number √2 lies strictly between 1 and 2. To refine the interval, square decimals: 1.4² = 1.96 and 1.5² = 2.25, so 1.4 < √2 < 1.5. Next, 1.41² = 1.9881 and 1.42² = 2.0164, giving 1.41 < √2 < 1.42. Squaring preserves order among nonnegative numbers, which is why these comparisons locate the root.

The interval endpoints and their squared values are exact numbers. A rounded display such as √2 ≈ 1.414 is different: ≈ means “approximately equal”, and 1.414² = 1.999396, not 2. The symbol √2 names the exact value even when we cannot write all its decimal digits. Keep the exact form during a calculation when possible, and choose a final approximation to suit the question’s required precision. A calculator display or screen position has finite precision.

Distinguish a fraction from a number no fraction can express

A rational number can be written exactly as m/n, where m and n are integers and n ≠ 0. Here m/n means m ÷ n. Examples include 1/8, −3/2 and 4 = 4/1; zero is rational because 0 = 0/1. In decimal form, rational numbers either terminate, such as 1/8 = 0.125, or eventually repeat a block, such as 1/3 = 0.333… with the digit 3 repeating forever.

An irrational number is a real number that cannot be written as a quotient of two integers with a nonzero denominator. √2 is an example: its decimal expansion continues forever without eventually repeating a fixed block. A proof of this fact requires an argument about all possible integer fractions; showing a few digits, or failing to see a pattern on a calculator, is not a proof. We will return to such proofs after developing fractions and divisibility.

A root symbol does not automatically signal an irrational number: √9 = 3 is rational. Conversely, a short exact name need not be a fraction or a finite decimal: √2 names one exact irrational value. The finite decimal 1.414 is rational, since 1.414 = 1414/1000. It approximates √2; it is not √2 itself.

The number line includes both kinds of real number

Once we choose zero, a positive direction and a unit interval, each real number corresponds to one position on the number line, and each position corresponds to a real number. The real numbers include all rational and all irrational numbers. Integers mark useful reference positions, but the spaces between integers also contain real numbers. For example, 0 < 1/2 < 1 < √2 < 2 < √5 < 3.

A finer scale helps us locate a number without changing it. Between 1.41 and 1.42, √2 has a definite exact position even if a drawing cannot show it exactly. Not every real number has a finite decimal representation: even the rational number 1/3 needs endlessly repeating digits, and irrational numbers have nonterminating, nonrepeating expansions. The drawn line, decimal labels and numerical bounds are representations of those positions, each with its own precision.

Check the inverse operation and resist misleading shortcuts

Check a proposed nonnegative square root by squaring it, and a proposed cube root by cubing it. For √25 = 5, verify 5² = 25 and 5 ≥ 0. For ∛(−27) = −3, verify (−3)³ = −27. If a decimal answer is approximate, its square or cube will generally be approximate too; compare its error with the required precision instead of expecting exact equality.

Square roots do not generally distribute over addition. For example, √(9 + 16) = √25 = 5, while √9 + √16 = 3 + 4 = 7. Powers do not generally distribute over addition either: (2 + 3)² = 25, while 2² + 3² = 13. Return to the meaning of the operation, keep parentheses and units, and check any proposed rule before using it.

Worked examples

Try it yourself

1. Find 2⁴, 2 × 4, (−3)², (−3)³ and −3². Explain why the powers and signs give different values.

   Show the answer and reasoning

   16, 8, 9, −27 and −9 respectively.

   2⁴ has four factors of 2, while 2 × 4 is one multiplication of two numbers. Two negative factors give a positive product, while three give a negative product. In −3², square the positive base 3 and then take its opposite.

2. Evaluate 5⁰, 2⁻² and 0³. Does the rule a⁰ = 1 determine 0⁰, and is 0⁻² defined here?

   Show the answer and reasoning

   1, 1/4 and 0. The stated zero-exponent rule excludes 0⁰, and 0⁻² is undefined in this arithmetic.

   5 is nonzero, so 5⁰ = 1. Also 2⁻² = 1 ÷ 2² = 1/4, and 0³ = 0 × 0 × 0 = 0. The rule a⁰ = 1 requires a ≠ 0; it cannot be extended to zero by dividing by the base. For 0⁻², the reciprocal would require 1 ÷ 0.

3. Rewrite 2² × 2³ as one power and evaluate it. Is 2² + 2³ equal to that power?

   Show the answer and reasoning

   2² × 2³ = 2⁵ = 32. The sum is 12, so it is different.

   Multiplication joins two factors of 2 with three factors of 2, giving five altogether. Addition combines the values 4 and 8 instead, so 2² + 2³ = 12. The exponent-sum rule is a rule for multiplication with the same base.

4. Find √25, every real solution of x² = 25, √((−5)²) and ∛(−27). Does √(−25) have a real value?

   Show the answer and reasoning

   5; x = 5 or x = −5; 5; and −3. √(−25) has no real value.

   The radical √25 chooses the nonnegative value, while both signs solve the squared equation. √((−5)²) = |−5| = 5. Cubing −3 gives −27, so its cube root is −3. No real square is negative.

5. A square has side length 4 cm, and a cube has edge length 4 cm. Find the area and volume with units. What happens to each if the length doubles?

   Show the answer and reasoning

   16 cm² and 64 cm³. Doubling the length gives 64 cm² and 512 cm³: four times the area and eight times the volume.

   Area multiplies two lengths, 4 × 4; volume multiplies three, 4 × 4 × 4. With length 8 cm, the corresponding products are 8² = 64 and 8³ = 512. Square and cubic units keep the kind of quantity visible.

6. Use 3.1² = 9.61 and 3.2² = 10.24 to locate √10. Would √10 = 3.1 be correct? Find an approximation whose error is guaranteed to be less than 0.05.

   Show the answer and reasoning

   3.1 < √10 < 3.2. The equality is false; the midpoint 3.15 is an approximation with error less than 0.05.

   The squared endpoints lie on opposite sides of 10, and squaring preserves order for nonnegative numbers. The root therefore lies strictly inside the interval. Its midpoint is 3.15, at distance 0.05 from either endpoint, so the root’s distance from the midpoint is smaller than 0.05. Write √10 ≈ 3.15, rather than an exact equality.

7. Classify 1/3, 1.414, √2 and √9 as rational or irrational. Then test the claim √(9 + 16) = √9 + √16.

   Show the answer and reasoning

   1/3, 1.414 and √9 are rational; √2 is irrational. The proposed root equality is false: its sides are 5 and 7.

   1/3 is already a quotient of integers, 1.414 = 1414/1000, and √9 = 3 = 3/1. √2 cannot be written as such a quotient; its finite decimal approximations are different rational numbers. Also √(9 + 16) = √25 = 5, whereas √9 + √16 = 3 + 4 = 7. A root does not generally distribute over addition.

What to carry forward

• A positive integer exponent counts equal factors, not an ordinary multiplication of the base by the exponent. Parentheses determine whether a sign belongs to the base.
• For a nonzero base, a⁰ = 1 and a⁻ⁿ = 1 ÷ aⁿ for positive integer n. The beginner rule leaves 0⁰ aside and does not define negative powers of zero.
• A principal square root is nonnegative. √(x²) = |x|, while ∛(x³) = x for real x. A negative input has no real square root but has a real cube root.
• Exact square comparisons locate a nonnegative root between exact bounds. A finite decimal or drawn position for an irrational root is approximate; use ≈ appropriately.
• Rational numbers are integer quotients with nonzero denominators; irrational numbers are the remaining real numbers. Both have exact positions on the real number line.
• Check powers and roots by their definitions and inverse operations, retain units, and test shortcuts rather than assuming roots or squares distribute over addition.