C4 Exponents and Radicals

Question 1

exponents

Answer 1

is the small superscripted number to the right of a larger number that tells you how many times to multiply the larger number, called the base

Question 2

exponential expressions

Answer 2

Consist of a base and a power. The general format is bⁿ where b is the base and n is the power. The base, b, has to be a positive number, and the power, n, is a real number.

Question 3

when the number x is involved in repeated multiplication of x times itself, then the number _____ can be used to describe how many multiplications are involved

Answer 3

when the number x is involved in repeated multiplication of x times itself, then the number n can be used to describe how many multiplications are involved

xⁿ = x · x · x ·x · · · n times

Question 4

the form for a number written in scientific notation is: N x 10^a. where the number N is a number between ____ and ____ and where a is an _____.

Answer 4

the form for a number written in scientific notation is: N x 10^a. where the number N is a number between 1 and 10 and where a is an integer.

Question 5

how do you write a number in scientific notation?

Answer 5

1. Determine where the decimal point is in the number and move it left or right until you have exactly one digit to the left of the decimal point.

This gives you a number between 1 and 10.

1. Count how many places (digits) you had to move the decimal point from its original position.

This is the absolute value of your exponent.

1. If you moved the original decimal point to the left your exponent is positive. If you moved it to the right it is negative.
2. Rewrite the number in scientific notation by making a product of your new, between-1and-10 number, times a 10 raised to the power of your exponent.

Question 6

why does scientific notation always use a decimal between 1 and 10?

Answer 6

Order of Magnitude: is a simple way to keep track of roughly how large or small a number is so you can compare numbers more easily.

Question 7

a geometric sequence is formed when each term is found by multiplying the previous term by a particular number, called the _____.

Answer 7

a geometric sequence is formed when each term is found by multiplying the previous term by a particular number, called the ratio.

Question 8

the sum of the terms of an infinite geometric sequence where the ratio, r , is a number between 0 and 1, is found by dividing the ____ ____of a sequence, a, by the _____ ____ 1 and r .

Answer 8

the sum of the terms of an infinite geometric sequence where the ratio, r , is a number between 0 and 1, is found by dividing the first term of a sequence, a, by the difference between 1 and r.

sum = a / 1 - r

• sum = 40 / 1 - 0.75 = 40 / 0.25 = 160*
• bouncing ball exmaple pg 62*

Question 9

how to multiply numbers with the same base?

Answer 9

add the exponents together

x^a · x^b = x^{a + b}

Question 10

when there is no exponent showing on a factor, you assume the exponent is ____.

Answer 10

when there is no exponent showing on a factor, you assume the exponent is 1 .

y = y¹

Question 11

you can add the exponents when multiplying numbers with the same base but you can multiply numbers that have the same power (in a mutiplication problem). The rule is that: _______

Answer 11

aⁿ · bⁿ = (a · b)ⁿ

4⁸ · 7⁸ = (4 x 7)⁸ = 28⁸

Question 12

how do you divide exponential expression with the same base?

Answer 12

you subtract the exponents and leave the base the same

x^a / x^b = x^a-b

Question 13

any number to the power of 0 equals ________________?

Answer 13

any number to the power of 0 equals 1 as long as the base number is not 0

a⁰ = 1

2⁴ / 2⁴ = 2^{4 - 4} = 2⁰ but correctly 2⁴ = 16 so 16 / 16 = 1

Question 14

the reciprocal of any number is the _____ ______ of the number. The product of a number and its reciprical is equal to _____.

Answer 14

the reciprocal of any number is the multiplicative inverse of the number. The product of a number and its reciprical is equal to 1

Question 15

the reciprocal of x^a is ____, which can also be written as x^-a. The variable x is any real number except ____, and a is any _____ _____. Also, to get rid of the negative exponent, you write x^-a = ____.

Answer 15

the reciprocal of x^a is 1 / x^{<u><strong>a</strong></u>} (as a fraction), which can also be written as x^-a. The variable x is any real number except 0, and a is any real real number. Also, to get rid of the negative exponent, you write x^-a​ = 1 / x^a .

Question 16

negative powers are a way of writing powers of _____ or ____ without using the _____ or _____.

Answer 16

negative powers are a way of writing powers of fractions or decimals without using the fractions or decimals.

Instead of writing (1/10)¹⁴ you can write 10^-14

Question 17

complex fraction

Answer 17

is a fraction with a fraction in it

Question 18

what is equal to 2^-3

Answer 18

= 1 / 2³ = 1/8

Question 19

what is the reciprocal of 6

Answer 19

is 1/6 or 6^-1

Question 20

(1/10)¹⁴ written differently

Answer 20

10^-14

Question 21

what is the formula to raise a power to a power

Answer 21

(xⁿ)^m = x^nm

in other words when the expression xⁿ is raised to the mth power, the new power of x is determined by multiplying n and m together

Question 22

what are the components of a root operation?

Answer 22

value under the radical = radicand

root power (exponent) = index

operator is the radical symbol √

Question 23

a radical is a non-binary operation that asks you, “what number times _____ gives you the number under the radical?” Another way of saying this is: “If √a = b, then ___ = a.”

Answer 23

a radical is a non-binary operation that asks you, “what number times itself gives you the number under the radical?” Another way of saying this is: “If √a = b, then b^{<u><strong>2</strong></u>} = a.”

Question 24

expressions with radicals can be multiplied or divided as long as the?

Answer 24

the root powers (index) are the same.

√2 x √3 = √2x3 = √6

³√8 / ³√4 = ³√8 /4 = ³√2

Question 25

expressions with radicals cannot be added or subtracted unless?

Answer 25

both the root power (index) and the value under the radical (radicand) are the same

√2 + √3 ≠ cannot be combined

√3 + √3 = 6

Question 26

the square root of 1 followed by an even number of zeros is always a 1 followed by?

Answer 26

half that many zeros

√100 = 10

√10,000 = 100

√1,000,000 = 1,000

Question 27

rules for changing a radical form to fractional exponents:

ⁿ√a = ____

ⁿ√a^m = ____

Answer 27

[in short divide the power of the radicand by the root power]

ⁿ√a = a^1/n : the nth root of “a” can be written as a fractional exponent with “a” raised to the reciprocal of that power.

ⁿ√a^m = a^m/n : when the nth root of a^m is taken, it’s raised to the 1/nth power

Using the “Powers of Powers” rule, the m and the 1/n are multiplied together

Question 28

the reciprocal of any number can be written by writing the same number as

Answer 28

a fraction of the two numbers and flipping the fraction

Example:

x = 3 and c = 4

x^c = 3⁴ and 1/x^c = 1/3⁴

3⁴ = 81 = 81/1

81/1 x 1/81 = 1

x^c * x^-c = 1

x^c * 1/x^c = 1

Question 29

even though x in the expression x² can be any real number and the n (2) can be any real number, they can both be ____ at the same time. ___ really has no meaning in algebra. Also if x is equal to ___ the n can’t be ____.

Answer 29

even though x in the expression x² can be any real number and the n (2) can be any real number, they can both be 0 at the same time. 0⁰ really has no meaning in algebra. Also if x is equal to 0 the n can’t be negative.