Powers and Square Roots

Raising a number to a power is a quick way to multiply a number by itself. For example, 2⁵, which you read as two to the fifth power, means that you multiply 2 by itself 5 times:

2⁵ = 2 x 2 x 2 x 2 x 2 = 32

The number 2 is called the base, and the number 5 is called the exponent.

Powers of ten — that is, powers with 10 in the base — are especially important because the number system is based on them. Fortunately, they’re very easy to work with. To raise 10 to the power of any positive whole number, write down the number 1 followed by the number of 0s indicated by the exponent. For example, 10³ is 1,000.

Here are some important rules for finding powers that contain 0 or 1:

• Every number raised to the power of 1 equals that number itself.

• Every number (except 0) raised to the power of 0 is equal to 1. For example, 10⁰ is 1 followed by no 0s — that is, 1.

• The number 0 raised to the power of any number (except 0) equals 0, because no matter how many times you multiply 0 by itself, the result is 0.

  Mathematicians have chosen to leave 0⁰ undefined — that is, it doesn’t equal any number.

• The number 1 raised to the power of any number equals 1, because no matter how many times you multiply 1 by itself, the result is 1.

When you multiply any number by itself, the result is a square number. So, when you raise any number to the power of 2, you’re squaring that number. For example, here’s 5², or five squared:

5² = 5 x 5 = 25

The inverse of squaring a number is called finding the square root of a number (inverse operations undo each other. When you find the square root of a number, you discover a new number which, when multiplied by itself, equals the number you started with. For example, here’s the square root of 25:

Sample questions

1. What is 3⁴?

   81. The expression 3⁴ tells you to multiply 3 by itself 4 times:

   3 x 3 x 3 x 3 = 81

2. What is 10⁶?

   1,000,000. Using the power of ten rule, 10⁶ is 1 followed by six 0s, so 10⁶ = 1,000,000.

3. What is the following?

   

   6. You want to find a number that, when multiplied by itself, equals 36. You know that 6 x 6 = 36, so

   

4. What is the following?

   

   16. You want to find a number that, when multiplied by itself, equals 256. Try guessing to narrow down the possibilities. Start by guessing 10:

   10 x 10 = 100

   256 > 100, so the answer is greater than 10. Guess 20:

   20 x 20 = 400

   256 < 400, so the answer is between 10 and 20. Guess 15:

   15 x 15 = 225

   256 > 225, so the answer is between 15 and 20. Guess 16:

   16 x 16 = 256

   This is correct, so

   

Practice questions

1. Find the value of the following powers:

   a. 6²

   b. 3⁵

   c. 2⁷

   d. 2⁸ (Hint: You can make your work easier by using the answer to c.)

2. Find the value of the following powers:

   a. 10⁴

   b. 10¹⁰

   c. 10¹⁵

   d. 10¹

Following are the answers to the practice questions:

1. Find the value of the following powers:

   a. 6² = 6 x 6 = 36.

   b. 3⁵ = 3 x 3 x 3 x 3 x 3 = 243.

   c. 2⁷ = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128.

   d. 2⁸ = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256. You already know from part c that 2⁷ = 128, so multiply this number by 2 to get your answer: 128 x 2 = 256.

2. Find the value of the following powers:

   a. 10⁴ = 10,000. Write 1 followed by four 0s.

   b. 10¹⁰ = 10,000,000,000. Write 1 followed by ten 0s.

   c. 10¹⁵ = 1,000,000,000,000,000. Write 1 followed by fifteen 0s.

   d. 10¹ = 10. Any number raised to the power of 1 is that number.