Radical Reasoning


You learned that multiplication is simply repeated addition. But did you know That there is an operation that substitutes for repeated multiplication?

Here is an example: 3x3x3x3x3 = 35 = 243. The 3 here is called the base, the 5 is the exponent, and the answer 243, is the power.

The Allowance Scam

You may have heard about the famed allowance scam that made a few lucky math-wiz kids into millionaires-at their parent’s expense. Here is how it works: Tell your parents that you are willing to

Accept two pennies for your allowance, but with one catch. Every day, the allowance should be raised to the next power. If your parents agree, how much money will your allowance be by the end of the month (in thirty days)?

Day 1 2 = 2
Day 2 2= 4
Day 3 2(9) = 8
Day 4 2(4) = 16

If you continue receiving your payments for thirty days, how much will you be entitled to receive on the last day of the month?

230 pennies = 1,073,741,824 pennies or $10.737,418.24-over ten million dollars!

A Calculator Tip

On many calculators, you use exponents by pressing a key marked x’. Put in the base number first. 3. Then press the xy key. Then press the exponent value. 5. Then press the key. You should see 243 as the answer. (A few calculators use the symbol instead of the x key)

A Radical Sign

You know that subtraction is the opposite of addition and division is the opposite of multiplication. The operation called squaring also has an opposite, called the square root. A square root of a number is written using a radical sign: 136 means “What number must I square (mul. tiply by itself) to get 36?” If you know your multiplication

Facts, you should be able to find an answer fairly easily it’s

6. Because 6 x 6 = 36. Now that you know what a square root means, finish filling in the following table.

The Square Root’s Double

Did you know that 36 has another square root in addition to 6? Can you think of what it is? What other two identical numbers can you multiply to get to 36? How about -6 -6? All positive numbers really have two square roots that are opposites of catch other, one positive and the other negative.


Square numbers are numbers that are the result of multiplying the same number twice. In other words, they can all be written with the exponent 2:

12 = 1, 22 = 4.32 = 9, 42 = 16, 52 = 25, 62 = 36,…

You can see what square numbers look like when you look at floor tiles, like the ones you probably have on the bathroom floor,

Drawkcab (Backward) Numbers

Palindromes are words that read the same forward and backward. “Racecar is a palindrome. Numbers can be palindromes if they are the same forward and backward, like 121. Here’s how to make your own palindrome numbers:

• Start with any number: 49 • Reverse it: 94 • Add the two: 49 + 94 – 143 • Reverse the sum: 341 • Add again:143 + 341 = 484

Presto! You’ve got a palindrome number: 484.

THREE IN A ROW his is a game for two players. The goal of the game is to identify SQUARE,

TRIANGULAR (divisible by 3), and PALINDROME numbers and be the first player to get three of your markers in a row. To play, you will need:

• A paper clip

A pencil Two sets of different place markers (such as coins, buttons, or small candies)

Use the point of a pencil to hold one end of a paper clip at the center of the spinner and spin the paper clip.

A Never Ending Jar an ancient Chinese porcelain jar has a sea, an island, two countries each with three mountains, and soon in Anno’s Mysterious Multiplying Jar by Masaichiro and Mitsumasa Anno (Philomel Books, 1983). See how factorials can riper sent the total number of mountains and much larger numbers!

Player 1 spins the paper clip and places a marker on a number in the grid that matches the category picked (for instance, if the paper clip lands on “square,” Player 1 can place a mark on 4, 9, 16, and so forth). Then, Player 2 does the same. Players take turns to see who can get three of their markers in a row first.

HINT: Some of the numbers fit more than one category.

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