SAT 2016

CHAPTER 9

Skill 3: Working with Exponentials and Radicals

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:

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 x^–n means the result when 1 is divided by x repeatedly n times.

In other words, .

Law #4: x^m × xⁿ = x^{m + n} (When multiplying exponentials with equal bases, add the exponents.)

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

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

Law #6: (When dividing exponentials with equal bases, subtract the exponents.)

Law #7: (When dividing exponentials with equal exponents, divide the bases.)

Law #8: (x^m)ⁿ = x^mn

Law #9:

Proof: This follows directly from Law #8. If we raise 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 x^a = x^b, then a = b.

Which of the following expressions is equivalent to ?

A)

B) 3n²

C) 3n⁶

D) 27n²

Therefore, the correct answer is (D).

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

A)

B) 3

C) 3ⁿ

D) 9²ⁿ

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.

The radical symbol () 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 ().

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

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

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

Law #3: (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.

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

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

and therefore

Which of the following is equivalent to (No calculator)

A)

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:

Substitute 18 = 9 × 2:

Apply Law #2:

Simplify :

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 means the principal, or non-negative square root, so 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).

If , what is the value of x?

A)

B)

C)

D)

(Hard)

Multiply by (x + 2):

Distribute:

Subtract :

Divide by :

Therefore, the correct answer is (D).

If , what is the value of k?

A) –3

B)

C)

D)

(Medium-hard)

Use Exponential Law #3:

Use Radical Law #1:

Substitute 4 = 2²:

Use Exponential Law #4:

If 2^a = 2^b, then a = b:

Multiply by –1:

If , what is the value of y³?

A)

B)

C)

D)

(Hard)

Use Radical Law #3:

Multiply by

Use Radical Law #1:

Use Exponential Law #4:

Divide by 3:

Square both sides:

Therefore, the correct answer is (A).

1

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

2

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

3

If w = –10³⁰, what is the value of

4

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

5

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

6

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

7

If , what is the value of ?

8

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

A)

B)

C)

D) 24

9

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

A)

B)

C) 2

D) 4

10

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

A) 2

B) 2ⁿ

C) 2^n–1

D) 2²ⁿ

11

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

A) 3^m+1

B) 3^2m

C) 3^3m

D) 3^3m+1

12

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

A) 5^2x

B) 5^2x+1

C) 5^3x

D) 25^2x

13

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

14

What is the smallest integer value of m such that

15

If , what is the value of k?

16

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

17

If , what is the value of n?

18

If , what is the value of n?

19

What is one possible value for x such that

20

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

A)

B)

C)

D)

21

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

A)

B)

C)

D)

22

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

A)

B)

C)

D)

23

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

A)

B)

C)

D)

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

Exponential Law #5:

Cancel common factors:

4. 510

5(2^2x) + 2^x

Exponential Law #8:

5(2^x)² + 2^x

Substitute 2^x = 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 2^–x could equal 2², 2⁴, or 2⁶. The greatest of these is 2⁶ = 64.

6. 80

FOIL:

Simplify:

Simplify:

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

7. 8

Cross-multiply:

Simplify:

ab = 9 – 5 = 4

Therefore ab^3/2 = 4^3/2 = 8

8. 5/3 or 1.66 or 1.67

9^x = 25

Substitute 9 = 3²:

(3²)^x = 25

Exponential Law #8:

3^2x = 25

Take square root:

3^x = 5

Divide by 3:

Exponential Law #6:

9. B

Simplify:

Simplify:

10. C

Cancel common factor:

Exponential Law #6:

2^{n - 1}

11. A

3^m + 3^m + 3^m

Combine like terms:

3(3^m)

Exponential Law #4:

3^m +¹

12. B

5y²

Substitute y = 5^x:

5(5^x)²

Exponential Law #8:

5(5^2x)

Exponential Law #4:

5^2x+1

Calculator

13. 64

Radical Law #1

n² = (64⁴)^1/2

Exponential Law #8:

n² = 64²

14. 5

Scientific Notation:

1 × 10^–m < 2.5 × 10^–5

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

15. 2.5

Exponential Law #6:

Simplify:

Express as exponentials:

Exponential Law #4:

3^k+1 = 3^3.5

Exponential Law #10:

k + 1 = 3.5

Subtract 1:

k = 2.5

16. 7

(x^m)³(x^m+1)² = x³⁷

Exponential Law #8:

(x^3m)(x^2m+2) = x³⁷

Exponential Law #4:

x^5m +² = x³⁷

Exponential Law #10:

5m + 2 = 37

Subtract 2:

5m = 35

Divide by 5:

m = 7

17. 6

Factor:

Divide by :

Simplify:

18 – 12 = 6 = n

18. 6

Substitute 8 = 2³:

Exponential Law #8:

Exponential Law #10:

Multiply by 12:

6 = n

19. 1 < x ≤ 1.56

Middle inequality:

Square both sides:

Divide by x:

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

Multiply by 25/16:

Last inequality:

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

Simplify:

Simplify:

Cancel common factor:

21. B Translate:

Square both sides:

x = 4x²

Divide by x:

22. D

Factor terms:

Cancel common factors:

Combine like terms:

23. B Pythagorean Theorem:

Simplify:

1 + x² = n

Subtract 1:

x² = n – 1

Take square root: