Deck 5: Exponents, Polynomials, and Polynomial Functions

Question 1

Simplify: $a ^ { - 3 } \cdot a ^ { - 4 } \cdot a ^ { 3 }$

A) $a ^ { 36 }$
B) $\frac { 1 } { a ^ { 4 } }$
C) $a ^ { 2 }$
D) $\frac { 3 } { a ^ { 4 } }$

Answer 1

$\frac { 1 } { a ^ { 4 } }$

Question 2

Simplify the expression. $\left( r ^ { 5 } s \right) ^ { 2 }$

A) $r ^ { 7 } s ^ { 3 }$
B) $\frac { r ^ { 10 } } { s ^ { 2 } }$
C) $r ^ { 5 } s ^ { 2 }$
D) $r ^ { 10 } s ^ { 3 }$
E) $r ^ { 10 } s ^ { 2 }$

Answer 2

$r ^ { 10 } s ^ { 2 }$

Question 3

Simplify the expression. Assume that the variable represents a nonzero real number. Write the answer using a positive exponent only. $7 ^ { 2 } r ^ { - 7 } \left( r ^ { 8 } \right) ^ { 5 }$

A) $\frac { 49 } { r ^ { 33 } }$
B) $49 r ^ { 6 }$
C) $7 r ^ { 6 }$
D) $7 r ^ { 47 }$
E) $49 r ^ { 33 }$

Answer 3

$49 r ^ { 33 }$

Question 4

Simplify the expression. Write the answer using positive exponents. $m ^ { - 4 } m ^ { 3 } m ^ { - 8 }$

A) $\frac { 1 } { m }$
B) $\frac { 1 } { 9 }$
C) $m ^ { 9 }$
D) $\frac { 1 } { m ^ { 9 } }$
E) $\frac { 1 } { m ^ { \frac { 1 } { 9 } } }$

Question 5

Simplify the expression. $\frac { 1 } { 4 ^ { - 3 } }$

A) -12
B) $- \frac { 1 } { 64 }$
C) -64
D) $\frac { 1 } { 64 }$
E) 64

Question 6

Simplify the expression. $\left( \frac { 2 a ^ { 3 } b ^ { 2 } z ^ { - 5 } } { 3 a ^ { - 4 } b ^ { - 2 } z ^ { 5 } } \right) ^ { - 3 }$

A) $\frac { 8 a ^ { 21 } b ^ { 12 } } { 27 z ^ { 30 } }$
B) $\frac { 27 a ^ { 30 } } { 8 z ^ { 21 } b ^ { 12 } }$
C) $\frac { 2 z ^ { 30 } } { 3 a ^ { 21 } b ^ { 12 } }$
D) $\frac { 27 b ^ { 30 } } { 8 a ^ { 21 } z ^ { 12 } }$
E) $\frac { 27 z ^ { 30 } } { 8 a ^ { 21 } b ^ { 12 } }$

Question 7

Simplify the expression. $x ^ { 5 } x ^ { 14 } x$

A) $x ^ { 20 }$
B) $x ^ { 19 }$
C) $x ^ { 18 }$
D) $x ^ { 22 }$
E) $x ^ { 21 }$

Question 8

Simplify the expression. $\left( \frac { 3 a ^ { 3 } b } { a ^ { - 2 } b ^ { 4 } } \right) ^ { 3 }$

A) $\frac { 27 a ^ { 15 } } { b ^ { 9 } }$
B) $\frac { 27 a ^ { 3 } } { b ^ { 9 } }$
C) $\frac { 27 } { a ^ { 3 } b ^ { 9 } }$
D) $\frac { 27 b ^ { 9 } } { a ^ { 15 } }$
E) $27 a ^ { 15 } b ^ { 9 }$

Question 9

Simplify the expression. $4 ^ { - 4 }$

A) -16
B) -256
C) 256
D) $- \frac { 1 } { 256 }$
E) $\frac { 1 } { 256 }$

Question 10

Simplify the expression. $\left( x ^ { 3 } \right) ^ { 7 }$

A) $x ^ { 10 }$
B) $\frac { 1 } { x ^ { 10 } }$
C) $x ^ { 4 }$
D) $\frac { 1 } { x ^ { 21 } }$
E) $x ^ { 21 }$

Question 11

Simplify the expression. $\left( \frac { a } { b ^ { 2 } } \right) ^ { 3 }$

A) $\frac { a ^ { 3 } } { b ^ { 6 } }$
B) $\frac { a ^ { 4 } } { b ^ { 5 } }$
C) $\frac { b ^ { 6 } } { a ^ { 3 } }$
D) $\frac { a ^ { 3 } } { b ^ { 5 } }$
E) $\frac { a ^ { 6 } } { b ^ { 3 } }$

Question 12

Simplify the expression. $\frac { \left( 2 x ^ { 4 } \right) ^ { - 2 } } { x ^ { 7 } x ^ { - 8 } x ^ { 0 } }$

A) $\frac { 1 } { 4 x ^ { 7 } }$
B) $- 4 x ^ { 7 }$
C) $\frac { 1 } { 4 x ^ { 5 } }$
D) $- \frac { 1 } { 4 x ^ { 7 } }$
E) $4 x ^ { 7 }$

Question 13

Simplify the expression. Assume that variables represent nonzero real numbers. Write the answer using positive exponents only. $\left( \frac { a ^ { 2 } } { b ^ { 3 } } \right) ^ { - 5 }$

A) $\frac { b ^ { 15 } } { a ^ { 10 } }$
B) $- \frac { a ^ { 10 } } { b ^ { 15 } }$
C) $\frac { a ^ { 10 } } { b ^ { 15 } }$
D) $- \frac { a ^ { - 10 } } { b ^ { - 15 } }$
E) $\frac { b ^ { 10 } } { a ^ { 15 } }$

Question 14

Simplify the expression. $\frac { 1 } { 6 ^ { - 2 } }$

A) -36
B) $- \frac { 1 } { 36 }$
C) 36
D) $- \frac { 1 } { 12 }$
E) $\frac { 1 } { 36 }$

Question 15

Simplify the expression. $( - 3 ) ^ { - 3 }$

A) $\frac { 1 } { 27 }$
B) -27
C) 27
D) $- \frac { 1 } { 27 }$
E) 9

Question 16

Simplify the expression. $- \frac { 5 t ^ { - 9 } t ^ { - 2 } } { t ^ { - 11 } }$

A) $- \frac { t } { 5 }$
B) $- 5 t$
C) $- \frac { 1 } { 5 }$
D) $- \frac { 5 } { t }$
E) -5

Question 17

Simplify: $\left( 2 x ^ { - 3 } \right) ^ { - 4 }$

A) $\frac { x ^ { 12 } } { 16 }$
B) $2 x ^ { 12 }$
C) $- 16 x ^ { 12 }$
D) $- 8 x ^ { 12 }$

Question 18

Simplify the expression. $\left( \frac { 2 } { 9 } \right) ^ { - 2 }$

A) $\frac { 81 } { 4 }$
B) $\frac { 9 } { 2 }$
C) $- \frac { 4 } { 81 }$
D) $\frac { 4 } { 81 }$
E) $- \frac { 81 } { 4 }$

Question 19

Simplify the expression. $\frac { c ^ { 12 } c ^ { 5 } } { c ^ { - 7 } }$

A) $c ^ { 10 }$
B) $\frac { 1 } { 24 }$
C) $\frac { 1 } { c ^ { 24 } }$
D) $\frac { 1 } { c ^ { 10 } }$
E) $c ^ { 24 }$

Question 20

Simplify the expression. $\left( b ^ { 8 } \right) ^ { 6 }$

A) $b ^ { 48 }$
B) $b ^ { 14 }$
C) $\frac { 1 } { b ^ { 48 } }$
D) $\frac { 1 } { b ^ { 2 } }$
E) $\frac { 1 } { b ^ { 14 } }$

Question 21

Simplify the expression. $x ^ { 5 } x ^ { 16 } x$

Question 22

Simplify the expression. $\left( x ^ { 3 } \right) ^ { 9 }$

Question 23

Simplify the expression. $\left( r ^ { 7 } s \right) ^ { 4 }$

Question 24

Simplify the expression. $\left( b ^ { 9 } \right) ^ { 4 }$

Question 25

Expressions such as $5 x ^ { 2 } \text { and } 6 ^ { 3 }$ are called __________ expressions.

Question 26

Simplify: $\left( \frac { 3 x y ^ { - 3 } } { 2 x ^ { - 4 } y ^ { 4 } } \right) ^ { 2 }$

A) $\frac { 9 } { 4 x ^ { 6 } y ^ { 14 } }$
B) $\frac { 3 x ^ { 10 } } { 2 y ^ { 14 } }$
C) $\frac { 9 x ^ { 10 } } { 4 y ^ { 14 } }$
D) $\frac { 4 } { 9 x ^ { 6 } y ^ { 9 } }$

Question 27

Simplify: $\frac { 5 x ^ { 0 } } { x ^ { - 3 } x ^ { - 4 } }$

A) 0
B) $\frac { x ^ { 7 } } { 5 }$
C) $\frac { 5 } { x ^ { 12 } }$
D) $5 x ^ { 7 }$

Question 28

Evaluate: $6 ^ { - 2 }$

A) -12
B) $\frac { 1 } { 36 }$
C) -36
D) 36

Question 29

In the exponential expression $x ^ { n }$ , x is called the _________.

Question 30

Simplify: $\frac { a ^ { - 2 } } { a ^ { - 14 } }$

A) $a ^ { 7 }$
B) $\frac { 1 } { a ^ { 16 } }$
C) $\frac { 1 } { a ^ { 12 } }$
D) $a ^ { 12 }$

Question 31

Simplify: $\frac { \left( 2 x ^ { - 3 } y \right) ^ { - 2 } } { x ^ { 4 } y ^ { - 6 } }$

A) $- 4 x ^ { 2 } y ^ { 4 }$
B) $\frac { x ^ { 2 } y ^ { 4 } } { 4 }$
C) $\frac { x ^ { 2 } } { 4 y ^ { 8 } }$
D) none of these

Question 32

Simplify the expression. Write the answer using positive exponents. $\frac { 1 } { 8 ^ { - 2 } }$

Question 33

Simplify: $\left( 4 x ^ { - 2 } y ^ { 5 } \right) ^ { - 2 }$

A) $\frac { 4 x ^ { 4 } } { y ^ { 10 } }$
B) $\frac { x ^ { 4 } } { 16 y ^ { 10 } }$
C) $\frac { - 16 x ^ { 4 } } { y ^ { 10 } }$
D) $\frac { - 8 x ^ { 4 } } { y ^ { 10 } }$

Question 34

Evaluate: $\left( \frac { 4 } { 5 } \right) ^ { - 3 }$

A) $\frac { 75 } { 48 }$
B) $\frac { 64 } { 125 }$
C) $- \frac { 64 } { 125 }$
D) $\frac { 125 } { 64 }$

Question 35

Evaluate: $( - 7 ) ^ { 0 }$

Question 36

Simplify the expression. Write the answer using positive exponents. $\frac { 1 } { 4 ^ { - 3 } }$

Question 37

Simplify the expression. Write the answer using positive exponents. $- \frac { 6 t ^ { - 11 } t ^ { - 2 } } { t ^ { - 13 } }$

Question 38

Simplify: $\left( a ^ { 8 } \right) ^ { 2 } \left( a ^ { 2 } \right) ^ { - 2 }$

A) $a ^ { 40 }$
B) $a ^ { 4 }$
C) $a ^ { 12 }$
D) $a ^ { 20 }$

Question 39

Simplify: $\left( a ^ { 4 } \right) ^ { - 2 } \cdot \left( 3 a ^ { 3 } \right) ^ { 3 }$

A) $27 a$
B) $\frac { 27 } { a }$
C) $9a$
D) 27

Question 40

Simplify: $x ^ { 6 } \cdot x ^ { 8 } \cdot x$

A) $3 x ^ { 15 }$
B) $x ^ { 48 }$
C) $x ^ { 15 }$
D) $x ^ { 14 }$

Question 41

Simplify the expression. Assume that variables represent nonzero real numbers. Write the answer using positive exponents only. $\left( \frac { a ^ { 5 } } { b ^ { 2 } } \right) ^ { - 4 }$

Question 42

Simplify the expression. Assume that the variable represents a nonzero real number. Write the answer using a positive exponent only. $2 ^ { 2 } r ^ { - 5 } \left( r ^ { 9 } \right) ^ { 2 }$

Question 43

Evaluate: $\left( 1 - \left( 1 - 7 ^ { - 1 } \right) ^ { - 1 } \right) ^ { - 1 }$

Question 44

Simplify the expression. $\left( \frac { a } { b ^ { 3 } } \right) ^ { 4 }$

Question 45

Determine what type of exponent must be used when writing real numbers between 0 and 1 using scientific notation.

A) negative integer
B) positive integer
C) 0

Question 46

Write the number in scientific notation. $0.0528 \times 10 ^ { 5 }$

A) $5.28 \times 10 ^ { 7 }$
B) $5.28 \times 10 ^ { - 2 }$
C) $5.28 \times 10 ^ { 2 }$
D) $52.8 \times 10 ^ { 4 }$
E) $5.28 \times 10 ^ { 3 }$

Question 47

Write the number in scientific notation. 3500

A) $0.35 \times 10 ^ { 4 }$
B) $35 \times 10 ^ { 2 }$
C) $3.5 \times 10 ^ { 3 }$
D) $3.5 \times 10 ^ { 2 }$
E) $3.5 \times 10 ^ { - 4 }$

Question 48

Write the number in scientific notation. $2000 \times 10 ^ { - 7 }$

A) $20.0 \times 10 ^ { - 3 }$
B) $2.0 \times 10 ^ { 4 }$
C) $0.2 \times 10 ^ { - 3 }$
D) $2.0 \times 10 ^ { - 4 }$
E) $2.0 \times 10 ^ { - 10 }$

Question 49

Write the number in scientific notation. $0.0314 \times 10 ^ { - 2 }$

A) $3.14 \times 10 ^ { 4 }$
B) $3.14 \times 10 ^ { - 4 }$
C) $31.4 \times 10 ^ { 3 }$
D) $0.314 \times 10 ^ { - 1 }$
E) $3.14 \times 10 ^ { - 2 }$

Question 50

Simplify the expression. Write the answer using positive exponents. $\frac { \left( 4 x ^ { 5 } \right) ^ { - 2 } } { x ^ { 3 } x ^ { - 10 } x ^ { 0 } }$

Question 51

Simplify the expression. Write the answer using positive exponents. $m ^ { - 3 } m ^ { 2 } m ^ { - 8 }$

Question 52

Simplify the expression. $\left( \frac { 2 } { 7 } \right) ^ { - 2 }$

Question 53

Write the number in scientific notation. $323 \times 10 ^ { 5 }$

A) $0.323 \times 10 ^ { 8 }$
B) $32.3 \times 10 ^ { 8 }$
C) $3.23 \times 10 ^ { 7 }$
D) $3.23 \times 10 ^ { 8 }$
E) $3.23 \times 10 ^ { 3 }$

Question 54

Write the number in scientific notation. 0.0073

A) $7.3 \times 10 ^ { 3 }$
B) $7.3 \times 10 ^ { - 2 }$
C) $7.3 \times 10 ^ { 4 }$
D) $7.3 \times 10 ^ { - 3 }$
E) $7.3 \times 10 ^ { - 4 }$

Question 55

Simplify the expression. Write the answer using positive exponents. $4 ^ { - 2 }$

Question 56

Simplify the expression. $( - 4 ) ^ { - 3 }$

Question 57

Simplify: $\frac { \left( 6 ^ { 5 m } \right) ^ { - 2 } \cdot 6 ^ { 3 m } } { \left( 6 ^ { 2 m } \right) ^ { - 7 } \cdot 6 ^ { 6 m } \cdot 6 ^ { - 2 m } }$ Assume there are no divisions by zero.

Question 58

Simplify the expression. Write the answer using positive exponents. $\frac { c ^ { 9 } c ^ { 4 } } { c ^ { - 11 } }$

Question 59

Simplify the expression. Write the answer using positive exponents. $\left( \frac { 3 a ^ { 2 } b } { a ^ { - 5 } b ^ { 4 } } \right) ^ { 3 }$

Question 60

Simplify the expression. Write the answer using positive exponents. $\left( \frac { 3 a ^ { 2 } b ^ { 4 } z ^ { - 3 } } { 2 a ^ { - 1 } b ^ { - 3 } z ^ { 5 } } \right) ^ { - 3 }$

Question 61

Write the number in standard notation. $8.5 \times 10 ^ { 2 }$

A) 0.85
B) 8500
C) 85
D) 850
E) 0.085

Question 62

The approximate number of atoms of carbon in a $\frac { 1 } { 2 }$ -carat diamond can be found by computing the following expression. $\frac { 6.0 \times 10 ^ { 23 } } { 1.2 \times 10 ^ { 2 } }$ Express the number of carbon atoms in scientific and in standard notation.

A) $5 \times 10 ^ { 22 } , 50,000,000,000,000,000,000$
B) $5 \times 10 ^ { 21 } , 5,000,000,000,000,000,000,000$
C) $5 \times 10 ^ { 21 } , 500,000,000,000,000,000,000$
D) $5 \times 10 ^ { 11 } , 500,000,000,000$
E) $5 \times 10 ^ { 19 } , 50,000,000,000,000,000,000$

Question 63

Write the numbers in scientific notation and perform the operations. Give the answer in scientific notation and in standard notation. $( 85,000,000,000 ) ( 4,700,000,000 )$

A) $3.995 \times 10 ^ { 19 } , \quad 39,950,000,000,000,000,000$
B) $0.3995 \times 10 ^ { 20 } , \quad 3,995,000,000,000,000,000$
C) $3.995 \times 10 ^ { 20 } , \quad 399,500,000,000,000,000,000$
D) $0.3995 \times 10 ^ { 19 } , \quad 3,995,000,000,000,000,000$
E) $3.995 \times 10 ^ { 18 } , \quad 3,995,000,000,000,000,000$

Question 64

Write 4,300,000 in scientific notation.

A) $4.3 \times 10 ^ { - 6 }$
B) $43 \times 10 ^ { 7 }$
C) $4.3 \times 10 ^ { 6 }$
D) $43 \times 10 ^ { - 7 }$

Question 65

Write 0.0015 in scientific notation.

A) $1.5 \times 10 ^ { - 4 }$
B) $15 \times 10 ^ { - 3 }$
C) $15 \times 10 ^ { - 2 }$
D) $15 \times 10 ^ { - 4 }$

Question 66

The mass of the Earth s oceans is only about $\frac { 1 } { 4400 }$ that of the Earth. If the mass of the Earth is $6.402 \times 10 ^ { 21 }$ tons, find the mass of the oceans.

A) $2.81688 \times 10 ^ { 22 }$ tons
B) $2.81688 \times 10 ^ { 21 }$ tons
C) $1.455 \times 10 ^ { 18 }$ tons
D) $1.455 \times 10 ^ { 19 }$ tons
E) $1.455 \times 10 ^ { 20 }$ tons

Question 67

Divide. Give the answer in scientific notation. $\frac { 2.25 \times 10 ^ { - 13 } } { 7.5 \times 10 ^ { - 6 } }$

A) $3 \times 10 ^ { - 9 }$
B) $3 \times 10 ^ { - 8 }$
C) $5 \times 10 ^ { - 7 }$
D) $7 \times 10 ^ { - 8 }$
E) $6 \times 10 ^ { - 8 }$

Question 68

Write $2.4 \times 10 ^ { - 4 }$ in standard notation.

A) 0.00024
B) 2,400
C) 0.0024
D) 24,000

Question 69

Multiply. Give the answer in scientific notation. $\left( 8.1 \times 10 ^ { 5 } \right) \left( 1.7 \times 10 ^ { 6 } \right)$

A) $1.343 \times 10 ^ { 12 }$
B) $1.394 \times 10 ^ { 12 }$
C) $1.377 \times 10 ^ { 12 }$
D) $1.377 \times 10 ^ { 11 }$
E) $1.343 \times 10 ^ { 11 }$

Question 70

Multiply. Give the answer in scientific notation. $\left( 8.9 \times 10 ^ { - 5 } \right) \left( 5.3 \times 10 ^ { 12 } \right)$

A) $4.717 \times 10 ^ { 7 }$
B) $4.717 \times 10 ^ { 8 }$
C) $5.057 \times 10 ^ { 7 }$
D) $\text { 4. } 547 \times 10 ^ { 7 }$
E) $4.887 \times 10 ^ { 8 }$

Question 71

Write the number in standard notation. $3.3 \times 10 ^ { - 8 }$

A) 330,000,000
B) 0.000000033
C) 0.0000000033
D) 33000000
E) 0.0033

Question 72

Perform the operation. Leave answer in scientific notation. $\frac { 1.44 \times 10 ^ { - 13 } } { 1.2 \times 10 ^ { 3 } }$

A) $1.2 \times 10 ^ { - 10 }$
B) $12 \times 10 ^ { - 10 }$
C) $1.2 \times 10 ^ { - 16 }$
D) $12 \times 10 ^ { - 16 }$

Question 73

Write $6.38 \times 10 ^ { 5 }$ in standard notation.

A) 0.0000638
B) 63,800
C) 638,000
D) 6,380,000

Question 74

Perform the operation. Give answer in scientific notation. $\left( 3.5 \times 10 ^ { 6 } \right) \left( 2.9 \times 10 ^ { 2 } \right)$

A) $10.15 \times 10 ^ { 8 }$
B) $1.015 \times 10 ^ { 10 }$
C) $1.015 \times 10 ^ { 9 }$
D) $1.015 \times 10 ^ { 8 }$

Question 75

Express the volume of the fish tank in scientific notation.

A) $9 \times 10 ^ { 9 } \mathrm {~mm} ^ { 3 }$
B) $1.8 \times 10 ^ { 10 } \mathrm {~mm} ^ { 3 }$
C) $9 \times 10 ^ { 10 } \mathrm {~mm} ^ { 3 }$
D) $9 \times 10 ^ { 12 } \mathrm {~mm} ^ { 3 }$
E) $4.5 \times 10 ^ { 9 } \mathrm {~mm} ^ { 3 }$

Question 76

Write the numbers in scientific notation and perform the operations. $\frac { 0.00000184 } { 0.0004 }$ Give the answer in scientific notation and in standard notation.

A) $4.6 \times 10 ^ { - 4 } , 0.00046$
B) $0.46 \times 10 ^ { - 3 } , 0.00046$
C) $0.46 \times 10 ^ { - 2 } , 0.0046$
D) $4.6 \times 10 ^ { - 2 } , 0.046$
E) $4.6 \times 10 ^ { - 3 } , 0.0046$

Question 77

Divide. Give the answer in scientific notation. $\frac { 6.3 \times 10 ^ { 3 } } { 3.5 \times 10 ^ { - 5 } }$

A) $1.8 \times 10 ^ { 8 }$
B) $1.8 \times 10 ^ { - 2 }$
C) $1.8 \times 10 ^ { 2 }$
D) $1.8 \times 10 ^ { - 8 }$

Question 78

Perform the operation. Leave answer in scientific notation. $\left( 3.2 \times 10 ^ { 6 } \right) \left( 4.4 \times 10 ^ { 2 } \right)$

A) $14.08 \times 10 ^ { 8 }$
B) $1.408 \times 10 ^ { 9 }$
C) $1.408 \times 10 ^ { 7 }$
D) $1.408 \times 10 ^ { 8 }$

Question 79

Write the number in standard notation. $4.64 \times 10 ^ { 0 }$

A) 0
B) 46.4
C) 0.464
D) 4.64
E) 1

Question 80

Write the numbers in scientific notation and perform the operations. Give the answer in scientific notation and in standard notation. $( 0.000000063 ) ( 3,800,000,000 )$

A) $2.394 \times 10 ^ { 3 } , 2,394$
B) $2.394 \times 10 ^ { 2 } , 239.4$
C) $0.2394 \times 10 ^ { 2 } , 23.94$
D) $2.394 \times 10 ^ { - 2 } , 0.02394$
E) $2.394 \times 10 ^ { 2 } , 0.02394$