SAT 2016

CHAPTER 9

THE SAT MATH: ADVANCED MATHEMATICS

Skill 3: Working with Exponentials and Radicals

Lesson 9: The Laws of Exponentials

When working with exponentials you must understand the Laws of Exponentials.

Law #1: If n is a positive integer, then xn means the result when 1 is multiplied by x repeatedly n times.

e.g.,     35 = 1 × 3 × 3 × 3 × 3 × 3 = 243

You might think that it’s unnecessary to include the 1 in this product, but including it will help clarify what zero, negative, and fractional exponents mean. For instance, think about the following sequence:

243, 81, 27, 9, 3, ___, ___, ___

What are the missing three terms in this sequence? With a little trial and error, you will see that the rule for getting each term is “divide the previous term by 3,” and therefore the missing terms are 1, 1/3, and 1/9. But notice, also, that these terms are just the descending integer powers of 3:

Images

And so on. If you explore this pattern, and patterns for the powers of other numbers, you will notice that some other laws clearly emerge.

Law #2: As long as x does not equal 0, x0 =1.

You can think of x0 as meaning “1 multiplied by x zero times, or not at all.” Therefore, the result is 1.

Law #3: If n is a positive integer, then xn means the result when 1 is divided by x repeatedly n times.

In other words, Images.

Images

Law #4: xm × xn = xm + n (When multiplying exponentials with equal basesadd the exponents.)

Images

Law #5: xn × yn =(xy)n (When multiplying exponentials with equal exponentsmultiply the bases.)

This law follows from the Commutative and Associative Laws of Addition.

Images

Law #6: Images (When dividing exponentials with equal basessubtract the exponents.)

Images

Law #7: Images (When dividing exponentials with equal exponentsdivide the bases.)

Images

Law #8: (xm)n = xmn

Images

Law #9: Images

Proof: This follows directly from Law #8. If we raise Images to the nth power, by Law #8 we must get x1 or x. The number that we must raise to the nth power in order to get x is, by definition, the “nth root of x.”

Law #10: If x >1 and xa = xb, then a = b.

Which of the following expressions is equivalent to Images?

A)   Images

B)   3n2

C)   3n6

D)   27n2

Images

Therefore, the correct answer is (D).

Which of the following expressions is equivalent to Images for all values of n?

A)   Images

B)   3

C)   3n

D)   92n

Images

Therefore, the correct answer is (B).

Alternately, we can plug in various values for n and find that the expression gives a value of n no ­matter what.

Lesson 10: The Laws of Radicals

The radical symbol (Images) is used to indicate roots, which are the inverse of exponentials. For instance, because 23 = 8, we can say that 2 is the “third root” or “cube root” of 8 (Images).

Law #9 of exponentials shows us that radicals (or “roots”) can be expressed as exponentials. For instance, Images. Therefore, we can use the Laws of Exponentials to simplify radical expressions.

Images

Law #1: Images (This is just the “reflected” ­version of Law of Exponentials #9.)

Law #2: Images (This follows directly from Law of Exponentials #5.)

Law #3: Images (This follows directly from Law of Exponentials #7.)

Working with square roots is much easier if you memorize the first 10 or so “perfect square integers”

22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49,

82 = 64, 92 = 81, 102 = 100, 112 = 121, 122 = 144. . .

This will help you both simplify and estimate radical expressions.

•   If the radicand has a perfect square factor, the radical can be simplified by factoring.

Images

•   If a fraction has a radical in the denominator, eliminate it by multiplying numerator and denominator by the radical.

Images

•   To estimate the value of square roots, notice which two consecutive perfect squares the radicand lies between.

Images and therefore Images

Which of the following is equivalent to Images (No calculator)

A)   Images

B)   7

C)   14

D)   19

(Medium) Notice that each answer choice is much simpler than the original expression. This suggests that the original expression can be simplified. Let’s begin by looking at the radical expressions. If you know your perfect squares you will see that neither radicand (the expression inside the radical) is a perfect square, but one of the radicands—18—is a multiple of a perfect square: 18 = 2 × 9.

Original expression: Images

Substitute 18 = 9 × 2: Images

Apply Law #2: Images

Simplify ImagesImages

Therefore, the correct answer is (C).

If x2 = 4, y2 = 9, and (x – 2)(y + 3) ≠ 0, what is the value of x + y?

A)   –5

B)   –1

C)   1

D)   5

Every positive number has two distinct square roots. For instance, both 5 and –5 are the square root of 25, because (5)2 = 25 and (–5)2 = 25.

However, the symbol Images means the principal, or non-negative square root, so Images and not –5.

(Easy) If x2 = 4, then x = ±2, and if y2 = 9, then y = ±2. But if (x – 2)(y + 3) ≠ 0, the x cannot equal 2 and y cannot equal –3. Therefore, x = –2 and y = 3, and x +y = 1, so the correct answer is (C).

Lesson 11: Solving radical and exponential equations

If Images, what is the value of x?

A)   Images

B)   Images

C)   Images

D)   Images

(Hard)

Images

Multiply by (x + 2):

Images

Distribute:

Images

Subtract Images:

Images

Divide by Images:

Images

Therefore, the correct answer is (D).

If Images, what is the value of k?

A)   –3

B)   Images

C)   Images

D)   Images

(Medium-hard)

Images

Use Exponential Law #3:

Images

Use Radical Law #1:

Images

Substitute 4 = 22:

Images

Use Exponential Law #4:

Images

If 2a = 2b, then a = b:

Images

Multiply by –1:

Images

If Images, what is the value of y3?

A)   Images

B)   Images

C)   Images

D)   Images

(Hard)

Images

Use Radical Law #3:

Images

Multiply by Images

Images

Use Radical Law #1:

Images

Use Exponential Law #4:

Images

Divide by 3:

Images

Square both sides:

Images

Therefore, the correct answer is (A).

Exercise Set 4 (No Calculator)

1

If 2a2 + 3a – 5a2 = 9, what is the value of a – a2?

Images

2

If (200)(4,000) = 8 × 10m, what is the value of m?

Images

3

If w = –1030, what is the value of Images

Images

4

If 2x = 10, what is the value of 5(22x) + 2x?

Images

5

If (x + 2)(x + 4)(x + 6) = 0, what is the greatest possible value of Images

Images

6

If Images, where a and b are integers, what is the value of a +b?

Images

7

If Images, what is the value of Images?

Images

8

If 9x = 25, what is the value of 3x–1?

A)   Images

B)   Images

C)   Images

D)   24

9

If Images and a and b are positive numbers, what is the value of Images

A)   Images

B)   Images

C)   2

D)   4

10

Which of the following is equivalent to Images for all positive values of n?

A)   2

B)   2n

C)   2n–1

D)   22n

11

Which of the following is equivalent to 3m + 3m + 3m for all positive values of m?

A)   3m+1

B)   32m

C)   33m

D)   33m+1

12

If x is a positive number and 5x = y, which of the ­following expresses 5y2 in terms of x?

A)   52x

B)   52x+1

C)   53x

D)   252x

Exercise Set 4 (Calculator)

13

If Images and n > 0, what is the value of n?

Images

14

What is the smallest integer value of m such that Images

Images

15

If Images, what is the value of k?

Images

16

If (xm)3(xm + 1)2 = x37 for all values of x, what is the value of m?

Images

17

If Images, what is the value of n?

Images

18

If Images, what is the value of n?

Images

19

What is one possible value for x such that Images

Images

20

Which of the following is equivalent to Images for all positive values of x?

A)   Images

B)   Images

C)   Images

D)   Images

21

The square root of a certain positive number is twice the number itself. What is the number?

A)   Images

B)   Images

C)   Images

D)   Images

22

Which of the following is equivalent to Images for all positive values of m and n?

A)   Images

B)   Images

C)   Images

D)   Images

23

Images

In the figure above, if n > 1, which of the following expresses x in terms of n?

A)   Images

B)   Images

C)   Images

D)   Images

EXERCISE SET 4 ANSWER KEY

No Calculator

1.  3

2a2 + 3a – 5a2 = 9

Simplify:

3a – 3a2 = 9

Divide by 3:

a – a2 = 3


2.  5   (200)(4,000) = 800,000 = 8 × 105


3.  1/8 or .125

Images

Exponential Law #5:

Images

Cancel common factors:

Images


4.  510

5(22x) + 2x

Exponential Law #8:

5(2x)2 + 2x

Substitute 2x = 10:

5(10)2 + 10

Simplify:

5(10)2 + 10 = 510


5.  64   If (x + 2)(x + 4)(x + 6) = 0, then x = –2, –4, or –6. Therefore 2x could equal 22, 24, or 26. The greatest of these is 26 = 64.


6.  80

Images

FOIL:

Images

Simplify:

Images

Simplify:

Images

Therefore a = 48 and b = 32 and a + b = 80.


7.  8

Images

Cross-multiply:

Images

Simplify:

ab = 9 – 5 = 4

Therefore ab3/2 = 43/2 = 8


8.  5/3 or 1.66 or 1.67

9x = 25

Substitute 9 = 32:

(32)x = 25

Exponential Law #8:

32x = 25

Take square root:

3x = 5

Divide by 3:

Images

Exponential Law #6:

Images


9.  B

Images

Simplify:

Images

Simplify:

Images


10.  C

Images

Cancel common factor:

Images

Exponential Law #6:

2n - 1


11.  A

3m + 3m + 3m

Combine like terms:

3(3m)

Exponential Law #4:

3m +1


12.  B

5y2

Substitute y = 5x:

5(5x)2

Exponential Law #8:

5(52x)

Exponential Law #4:

52x+1

Calculator

13.  64

Images

Radical Law #1

n2 = (644)1/2

Exponential Law #8:

n2 = 642


14.  5

Images

Scientific Notation:

1 × 10m < 2.5 × 10–5

Substitution and checking makes it clear that m = 5 is the smallest integer that satisfies the inequality.


15.  2.5

Images

Exponential Law #6:

Images

Simplify:

Images

Express as exponentials:

Images

Exponential Law #4:

3k+1 = 33.5

Exponential Law #10:

k + 1 = 3.5

Subtract 1:

k = 2.5


16.  7

(xm)3(xm+1)2 = x37

Exponential Law #8:

(x3m)(x2m+2) = x37

Exponential Law #4:

x5m +2 = x37

Exponential Law #10:

5m + 2 = 37

Subtract 2:

5m = 35

Divide by 5:

m = 7


17.  6

Images

Factor:

Images

Divide by Images:

Images

Simplify:

18 – 12 = 6 = n


18.  6

Images

Substitute 8 = 23:

Images

Exponential Law #8:

Images

Exponential Law #10:

Images

Multiply by 12:

6 = n


19.  1 < x ≤ 1.56

Images

Middle inequality:

Images

Square both sides:

Images

Divide by x:

Images

(Since x > 0, we do not “swap” the inequality.)

Multiply by 25/16:

Images

Last inequality:

Images

Square both sides:

x < x2

Divide by x:

1 < x

Therefore, x must be both greater than 1 and less than or equal to 1.56.


20.  B

Images

Simplify:

Images

Simplify:

Images

Cancel common factor:

Images


21.  B   Translate:

Images

Square both sides:

x = 4x2

Divide by x:

Images


22.  D

Images

Factor terms:

Images

Cancel common factors:

Images

Combine like terms:

Images


23.  B   Pythagorean Theorem:

Images

Simplify:

1 + x2 = n

Subtract 1:

x2 = n – 1

Take square root:

Images