## SAT SUBJECT TEST MATH LEVEL 1

## TOPICS IN ARITHMETIC

## CHAPTER 2

Basic Arithmetic

### SQUARES AND SQUARE ROOTS

The exponent that appears most often on the Math 1 test is 2. Although *a*^{2} can be read “*a* to the second,” it is usually read “*a* squared.” You will see 2 as an exponent in many formulas. For example:

*• A* = *s* ^{2 }(the area of a square)

*• A* = π*r* ^{2 }(the area of a circle)

*• a*^{2} + *b*^{2} = *c* ^{2 }(the Pythagorean theorem)

*• x*^{2} - *y* ^{2 }= (*x* - *y*)(*x* + *y*) (factoring the difference of two squares)

Numbers that are the squares of integers are called ** perfect squares**. You should recognize at least the squares of the integers from 0 through 15.

Of course if you need to evaluate 13^{2}, you can use your calculator. However, it is often helpful to recognize these perfect squares. That way, if you see 169, you will immediately think “that is 13^{2}.”

Two numbers, 5 and –5, satisfy the equation *x*^{2 }= 25. The positive one, 5, is called the ** square root** of 25 and is denoted by the symbol . Clearly, each perfect square has a square root: , , , and . However, it is an important fact that

*every positive number*has a square root.

**Key Fact A12**

**For any positive**

**number**

*a*, there is a positive number*b***that satisfies the equation**=

*b*^{2}

*a*. That number,*b*, is called the square root of*a***and is written**

**. So, for any positive number**, .

*a***Key Fact A13**

**For any positive numbers a**

**and**

*b*:For example, and .

**Don”t Get Confused**

•

For example:

•

•

•

The expression *a*^{3} is often read “*a* cubed.” Numbers that are the cubes of integers are called ** perfect cubes**. You should memorize the perfect cubes in the following table.

The only other powers you should recognize immediately are the powers of 2 up to 2^{10}.

In the same way that we write to indicate that *b*^{2} = *a*:

• We write to indicate that *b*^{3} = *a* and call *b* the *cube root* of *a*.

• We write to indicate that *b*^{4} = *a* and call *b* the fourth root of *a*.

• For any integer *n* 2, we write to indicate that *b ^{n}* =

*a*and call

*b*the

*n*th root of

*a*.

For example:

• because 5^{3} = 125.

• because (–2)^{3 }= –8.

• because 2^{10} = 1024.

Note that is undefined because there is no real number *x* such that *x* ^{2 }= –64. If you enter on your calculator, you will get an error message. (In Chapter 17, you will read about the imaginary unit *i* and will review the fact that is 8*i*.)

**Key Fact A14**

**For any real number a**

**and integer**

*n***2:**

• **If n**

**is odd, then**

**is the unique real number**

*x***that satisfies the equation**=

*x*^{n}

*a.*• **If n**

**is even and**

*a***is positive, then**

**is the unique**

*positive***number**

*x***that satisfies the equation**=

*x*^{n}

*a.*We can now expand our definition of exponents to include fractions.

**Key Fact A15**

**For any positive number b**

**and positive integers**

*n***and**

*m***with**

*n***2:**

•

•

For example:

The laws of exponents, listed in KEY FACT A11, are equally valid if any of the exponents are fractions. For example: