Jugami Pre-Algebra

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Exponents and Square Roots

A short way to write repeated multiplication, and how to undo a square.

A chessboard has 8 × 8 = 64 squares. A Rubik’s Cube is 3 × 3 × 3. Writing the same factor again and again gets long, so mathematicians use an exponent: 8 × 8 is 8², and 3 × 3 × 3 is 3³.

This page covers powers, squares and cubes as pictures, powers of 10, the zero exponent, exponents in the order of operations, and square roots, including how to estimate one. Then try the practice problems and the Power Up game.

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A square made of 3 by 3 unit squares, showing 3 squared is 9, next to a cube made of 3 by 3 by 3 unit cubes, showing 3 cubed is 27 3² = 3 × 3 = 99 unit squares333³ = 3 × 3 × 3 = 2727 unit cubes

Powers: base and exponent

In 2⁵ the big number 2 is the base and the small raised 5 is the exponent. The exponent tells how many times the base is used as a factor: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. The whole expression is called a power; read 2⁵ as “2 to the fifth power” or “2 to the fifth.” Any number to the first power is itself: 7¹ = 7.

A common mistake is to multiply the base by the exponent. 2⁵ is not 2 × 5 = 10; it is 32.

Example 1: Evaluate a power

Find 3⁴, and write 5 × 5 × 5 as a power.

  1. 3⁴ means four 3s multiplied: 3 × 3 × 3 × 3.
  2. Work left to right: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81.
  3. 5 × 5 × 5 uses 5 as a factor three times, so it is 5³ (which equals 125).

Squares and cubes

The second power is called the square because a square with side 3 has an area of 3 × 3 = 3² = 9 square units. The third power is called the cube because a cube with edge 3 is built from 3 × 3 × 3 = 3³ = 27 unit cubes. That is why 3² is read “3 squared” and 3³ is read “3 cubed.”

Example 2: Area and volume

A square tile is 6 inches on each side. A box is a cube 4 feet on each edge. Find the tile’s area and the box’s volume.

  1. Area of the square: 6² = 6 × 6 = 36 square inches.
  2. Volume of the cube: 4³ = 4 × 4 × 4 = 64 cubic feet.
  3. The units are squared or cubed too: square inches for area, cubic feet for volume.

Powers of 10

Powers of 10 are easy: 10ⁿ is 1 followed by n zeros. 10² = 100, 10³ = 1,000, and 10⁶ = 1,000,000 (one million). They are the place values of our number system, so any whole number can be written with them. Scientists use them for very big numbers: the Sun is about 1.5 × 10⁸ kilometers from Earth.

A place-value chart for 4,352. The thousands place is 10 cubed, or 1,000; hundreds is 10 squared, or 100; tens is 10 to the first, or 10; ones is 10 to the zero, or 1 Each place is a power of 1010³thousands1,000410²hundreds100310¹tens10510⁰ones124,352 = 4×10³ + 3×10² + 5×10¹ + 2×10⁰Moving one place left multiplies by 10
Each place value is a power of 10. The exponent counts how many places the digit is to the left of the ones place.

Example 3: Expanded form with powers of 10

Write 7 × 10⁴ in standard form, and write 3,060 using powers of 10.

  1. 10⁴ = 10,000, so 7 × 10⁴ = 70,000.
  2. 3,060 has 3 thousands, 0 hundreds, 6 tens and 0 ones.
  3. So 3,060 = 3 × 10³ + 6 × 10¹.

The zero exponent

What could 2⁰ mean? You can’t use 2 as a factor zero times and get a picture, but a pattern answers the question. Each time the exponent goes down by 1, the value is divided by 2: 16, 8, 4, 2. The next step is 2 ÷ 2 = 1, so mathematicians agree that 2⁰ = 1. The same reasoning works for every nonzero base: 5⁰ = 1, 10⁰ = 1, 99⁰ = 1. (0⁰ is a special case you will meet later; on this page every base is nonzero.)

Bars drawn to scale for 2 to the 4th (16), 2 cubed (8), 2 squared (4), 2 to the first (2) and 2 to the zero (1). Each bar is half the one before, so the pattern gives 2 to the zero equals 1 Divide by 2 each step162⁴÷282³÷242²÷222¹÷212⁰The pattern says 2⁰ = 1
Going down the powers of 2, each bar is half as tall as the one before. Halving 2 gives 1, which is 2⁰.

Exponents in the order of operations

When an expression mixes operations, work in this order: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. An exponent belongs only to the number (or parentheses) directly before it: in 2 × 3², only the 3 is squared.

Example 4: Exponents before multiplying

Evaluate 5 + 2 × 3².

  1. Exponent first: 3² = 9.
  2. Then multiply: 2 × 9 = 18.
  3. Then add: 5 + 18 = 23. (Adding first would give 7 × 9 = 63, which is wrong.)

Example 5: Parentheses change the answer

Compare (2 + 3)² with 2² + 3².

  1. (2 + 3)²: parentheses first, 2 + 3 = 5, then 5² = 25.
  2. 2² + 3²: exponents first, 4 + 9 = 13.
  3. They are different, so you can’t “share” an exponent over addition.

Perfect squares and square roots

A perfect square is a whole number times itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on. The square root undoes squaring. √49 asks, “what number times itself is 49?” The answer is 7, because 7² = 49. In pictures, the square root of an area is the side of the square. The symbol √ is called the radical sign. (−7 × −7 is also 49, but √49 means the positive root, 7.)

Squares of side 1, 2, 3, 4 and 5 units, made of 1, 4, 9, 16 and 25 unit squares, drawn to the same scale The first five perfect squares1²=12²=43²=94²=165²=25√25 = 5 because a square with area 25 has side 5
The first five perfect squares, all at the same scale. The side of each square is the square root of its area.

Example 6: Square roots

Find √144 and √(9 + 16).

  1. 12 × 12 = 144, so √144 = 12.
  2. The radical sign groups like parentheses, so add first: 9 + 16 = 25.
  3. √25 = 5. (Note √9 + √16 = 3 + 4 = 7 is different.)

Estimating square roots

Most whole numbers are not perfect squares, so their square roots are not whole numbers. You can still trap a square root between two whole numbers by finding the perfect squares just below and just above it. Since 49 < 50 < 64, √50 is between √49 = 7 and √64 = 8.

A number line from 7 to 8 in tenths. 7 squared is 49 and 8 squared is 64, and halfway, 7.5 squared is 56.25. The square root of 50, about 7.07, sits just past 7 Where is √50?77² = 4988² = 647.57.5² = 56.25√50 ≈ 7.0749 < 50 < 56.25, so √50 is between 7 and 7.5
√50 is a little more than 7. Checking the halfway point, 7.5² = 56.25, shows it is closer to 7 than to 8.

Example 7: Between which two whole numbers?

Between which two whole numbers is √90? Which one is it closer to?

  1. Perfect squares near 90: 9² = 81 and 10² = 100.
  2. 81 < 90 < 100, so √90 is between 9 and 10.
  3. Halfway is 9.5, and 9.5² = 90.25. 90 is just under 90.25, so √90 is a little closer to 9. (A calculator gives about 9.49.)

How do we know?

Why is 10³ a 1 with three zeros? Each factor of 10 shifts every digit one place to the left and adds a zero: 1 × 10 = 10, × 10 = 100, × 10 = 1,000. Three factors, three zeros.

Why must 2⁰ be 1? Going from 2³ to 2² removes one factor of 2, which divides by 2. Doing that once more from 2¹ = 2 leaves 2 ÷ 2 = 1. Any other value would break the pattern, and rules such as 2³ × 2⁰ = 2³⁺⁰ = 2³ only work if 2⁰ = 1.

Why does the halfway check work? Squaring keeps order for positive numbers: if a < b then a² < b². So √50 < 7.5 exactly when 50 < 7.5² = 56.25.

History: small raised numbers

People worked with square roots long before there were symbols for them. A small Babylonian clay tablet called YBC 7289, probably from between 1800 and 1600 BC, shows a square with its diagonals and gives the square root of 2 in base 60, accurate to about six decimal digits.

The √ sign came much later. The German mathematician Christoff Rudolff’s algebra book Coss (1525) is the first known printed use of √ for square roots (Wikipedia). In 1637 René Descartes introduced the raised exponent, as in a³, in La Géométrie, and Wikipedia says he was also the first to join the √ sign to a bar over the number, the radical sign we use today. Descartes himself often still wrote squares the long way, as xx.

Practice problems

Try each one, then open it to compare.

1. Evaluate 2⁶.

2 × 2 × 2 × 2 × 2 × 2 = 64.

2. Write 9 × 9 × 9 × 9 as a power.

Four factors of 9: 9⁴.

3. A cube has edges 5 cm long. What is its volume?

5³ = 5 × 5 × 5 = 125 cubic centimeters.

4. Write 6 × 10⁵ in standard form.

10⁵ = 100,000, so 6 × 10⁵ = 600,000.

5. Evaluate 8⁰ + 8¹.

8⁰ = 1 and 8¹ = 8, so the sum is 9.

6. Evaluate 20 − 3 × 2².

2² = 4, 3 × 4 = 12, 20 − 12 = 8.

7. Find √121.

11 × 11 = 121, so √121 = 11.

8. Between which two whole numbers is √30? Which is it closer to?

5² = 25 and 6² = 36, so it is between 5 and 6. 5.5² = 30.25 and 30 < 30.25, so it is closer to 5.

Game: Power Up

Ten problems: powers, zero exponents, powers of 10, order of operations, square roots, and estimating a square root to the nearest whole number. Type a whole number such as 81 or 1000 (commas like 1,000 are fine). Press Enter. One point per correct answer. Your best score is saved only on this device. About 4 minutes.

Talk About It

  • A friend says 3⁴ = 12. What did they do, and how would you explain the right answer?
  • Why do you think 10² and 10³ are called “squared” and “cubed” but 10⁴ has no special name?
  • Is (3 + 4)² the same as 3² + 4²? Draw a square to show why or why not.
  • Without a calculator, how would you decide whether √70 is closer to 8 or to 9?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami.