Unit 7 Review

Unit 7 Review

Part 1

Exponent Rules

A Quick Review

Zero Rule

Any nonzero base raised to the power of zero is 1

Examples:

•  $10^0=1$  

•  $\left(200,000,000,000,000\right)^0=1$  

•  $\left(-356\right)^0=1$  

•  $a^0=1,\ if\ a\ \ne0$  

Multiple Choice

 $5^0=?$  

• 1

• 5

• 0

• 50

Multiple Choice

 $\left(-723\right)^0=?$  

• -723

• 0

• 1

• -1

Multiple Choice

 $-34^0=?$  
BE CAREFUL!

• -1

• 1

• 0

• -34

 Be mindful of what your base is!

 $\left(-a\right)^x$  The base is  $-a$  

 $-a^x$  The base is just a, we can think of it as $-1\ \cdot\ a^x$  

• Keep this in mind!

Product rule:

When multiplying two powers that have the same base, ADD the exponents!

Examples:

•  $2^2\cdot2^3\ =2^{2+3}\ =2^5$  

•  $5^{-3}\cdot5^{10}=5^{-3+10}=5^7$  

•  $a^3+a^{15}=a^{18}$  

•  $a^m+a^n=a^{m+n}$  

Multiple Choice

 $3^3\cdot3^{10}=?$  

•  $3^{30}$  

•  $3^7$  

•  $3^{13}$  

•  $3^{\frac{1}{3}}$  

Multiple Choice

 $5^0\cdot5^{15}=?$  

•  $5^0$  

•  $5^{15}$  

•  $5^1$  

•  $1$  

Multiple Choice

 $a^{-4}\cdot a^{10}=?$  

•  $a^6$  

•  $a^{14}$  

•  $a^{-40}$  

•  $a^{-6}$  

Quotient Rule
When dividing two powers that have the same base, SUBTRACT the exponents!
Examples:

•  $\frac{10^8}{10^2}=10^{8-2}=10^6$  

•  $\frac{5^{15}}{5^3\ }=5^{15-3}=5^{12}$  

•  $\frac{a^{12}}{a^5}=a^{12-5}=a^7$  

•  $\frac{a^m}{a^n}=a^{m-n}$  

Multiple Choice

 $\frac{15^{100}}{15^{20}}=?$  

•  $15^5$  

•  $15^{2000}$  

•  $15^{120}$  

•  $15^{80}$  

Multiple Choice

 $\frac{100^8}{100^0}=?$  

•  $100^8$  

•  $100^0$  

•  $100^1$  

•  $100^{-8}$  

Multiple Choice

 $\frac{a^{10}}{a^{-5}}=?$  

•  $a^5$  

•  $a^{15}$  

•  $a^{-50}$  

•  $a^{50}$  

Power Raised to a Power

When you raise a power to a power, just MULTIPLY the exponents!

Examples:

•  $\left(4^2\right)^3=4^{2\cdot3}=4^6$  

•  $\left(200^3\right)^{10}=200^{3\cdot10}=200^{30}$  

•  $\left(a^{-2}\right)^3=a^{-2\cdot3}=a^{-6}$  

•  $\left(a^m\right)^n=a^{m\cdot n}$  

Multiple Choice

 $\left(4^2\right)^3=?$  

•  $4^5$  

•  $4^{-1}$  

•  $4^6$  

•  $4^1$  

Multiple Choice

 $\left(15^5\right)^0=?$  

•  $1$  

•  $15^5$  

•  $undefined$  

•  $15^{-5}$  

•  $0$  

Multiple Choice

 $\left(a^{-2}\right)^{-6}$  

•  $a^{12}$  

•  $a^{-12}$  

•  $a^4$  

•  $a^{-8}$  

Negative Exponents

**If there is a negative exponent in the NUMERATOR: keep the base, make the exponent positive, and move it to the denominator

(If there's a negative exponent in the DENOMINATOR: keep the base, make the exponent positive, and move it to the numerator)

EXAMPLES:

•  $10^{-3}=\frac{1}{10^3}$  

•  $5^{-20}=\frac{1}{5^{20}}$  

•  $a^{-m}=\frac{1}{a^m}$  

Multiple Choice

 $71^{-2}=?$  

•  $71^2$  

•  $\frac{1}{71^2}$  

•  $-71^2$  

•  $\frac{71^2}{1}$  

Multiple Choice

 $a^{-3}=?$  

•  $a^3$  

•  $-a^3$  

•  $\frac{1}{a^3}$  

•  $\frac{1}{a^{-3}}$  

Multiple Choice

Challenge!
 $\left(a^{-6}\right)^2=?$  

•  $\frac{1}{a^{12}}$  

•  $\frac{1}{a^4}$  

•  $\frac{1}{a^3}$  

•  $\frac{1}{a^8}$  

Multiplying Powers With different bases, but the SAME EXPONENT:
   -Multiply the bases
   -Keep the Power

Examples:

•  $2^3\cdot5^3=\left(2\cdot5\right)^3=10^3$  

•  $6^8\cdot5^8=\left(6\cdot5\right)^8=30^8$  

•  $10^a\cdot3^a=\left(10\cdot3\right)^a=30^a$  

•  $a^m\cdot b^m=\left(a\cdot b\right)^m$  

Multiple Choice

 $9^2\cdot3^2=?$  

•  $27^4$  

•  $27^2$  

•  $12^2$  

•  $12^4$  

Multiple Choice

 $5^{-2}\cdot3^{-2}=?$  

•  $15^{-2}$  

•  $15^{-4}$  

•  $8^{-2}$  

•  $8^{-4}$  

Multiple Select

 $Select\ all\ the\ equations\ that\ are\ EQUAL\ to\ 10^3$  

•  $10^2\cdot\ 10^1$  

•  $\left(10^3\right)^1$  

•  $10^{-3}$  

•  $\frac{10^5}{10^2}$  

Multiple Select

Select all the expressions that are equal to  $7^{-8}$  

•  $\frac{1}{7^8}$  

•  $\frac{7^1}{7^9}$  

•  $\left(7^2\right)\left(7^{-4}\right)$  

•  $\frac{7^{15}}{7^7}$  

Multiple Choice

 $\frac{\left(5^{12}\cdot5^7\right)}{\left(5^3\cdot5^7\right)}=?$  

What is the above written as with a single, positive exponent?

• 0

•  $5$  

•  $5^{63}$  

•  $5^9$  

Open Ended

Place a number in each box so that:

 Each equation is true
AND
Each equation has at least one negative number.

Write your answer as "A=_ and B=_"

(hint-click the image to make it bigger)

Open Ended

Place a number in each box so that:

 Each equation is true
AND
Each equation has at least one negative number.

Write your answer as "A=_ and B=_"

(hint-click the image to make it bigger)

Open Ended

Place a number in each box so that:

 Each equation is true
AND
Each equation has at least one negative number.

Write your answer as "A=_ and B=_"

(hint-click the image to make it bigger)

Part 2:

Scientific Notation

How do we write something in scientific notation?

Written as two factors:

• The first factor is a number that is greater than or equal to 1, but less than 10<

• The second factor is a power of 10

•  $1\le a<10\ \times10^n$  

Some examples...

 $3.14\ \times10^5$  
 $9.99\ \times\ 10^2$  
 $1.9\ \times10^{11}$  
 $5.8\times10^{-3}$  
 $3.8\times10^{-10}$  

Multiple Choice

 $1.8\times10^2$  

Is the above written in scientific notation?

• Yes

• No

Multiple Choice

 $0.86\ \times10^{20}$  
Is the above written in scientific notation?

• Yes

• No

Multiple Choice

 $110\times10^{13}$  
Is the above written in scientific notation?

• Yes

• No

From Scientific Notation to Standard Form

• The power of 10 gives us information on how many decimal places to move the decimal point when evaluating the expression.

• If the exponent is POSITIVE the decimal place is moved to the RIGHT

• If the exponent is NEGATIVE the exponent is moved to the LEFT

• Example:

•  $3.24\times10^5=324000$  move 5 to the right

•  $1.27\times10^{-3}=0.00127$  move the decimal place 3 to the left

Multiple Choice

What is the value of: $2.28\ \times10^2$  

• 0.228

• 22.8

• 0.0228

• 228

Multiple Choice

What is the value of: $3\times10^{-4}$  

• 0.0003

• 30000

• 300

• .00003

Writing From Standard to Scientific Notation

• Move the decimal place so the value is greater than or equal to one but less than 10

• From the new decimal place, COUNT how many times it is moving to the original, this is your exponent.

• If you are counting to the right the exponent is POSITIVE

• if you are counting to the left the exponent is NEGATIVE

Example: 412,000 and 0.00015

 $4.12\times10^5$  and  $1.5\ \times10^{-4}$  

Multiple Choice

Write the following using scientific notation: 325

•  $3.25\times10^2$  

•  $3.25\ \times10^{-2}$  

•  $325\times10^2$  

•  $325\times10^{-2}$  

Multiple Choice

Write the following using scientific notation: 6,700,000

• $6.7\times10^6$

• $6.7\times10^5$
  

• $6.7\times10^{-6}$

• $6.7\times10^{-5}$

Multiple Choice

Write the following using scientific  notation: 0.0028

•  $2.8\times10^{-3}$  

•  $2.8\times10^{-4}$  

•  $2.8\times10^3$  

•  $2.8\times10^4$  

Part 3: A deeper look

What are the points on the two ends of the expanded number line? What are the ticks in between?

Multiple Choice

What number is represented by point P?

• $5.5\times10^{-3}$

• $5.6\times10^{-3}$

• $5.7\times10^{-3}$

• $5.8\times10^{-3}$

When dividing two expressions that are in scientific notation, split the factors and divide each one separately

(it makes it easier!)

Multiple Choice

There are about $8.6\times10^5$ people in a particular coastal city. There are about  $4.3\times10^3$  dogs in that city. About how many times more people live in the city than dogs?

• 2 times as many

• 20 times as many

• 200 times as many

• 2,000 times as many

When multiplying two scientific notation expressions together:

• Group the numbers together and multiply

• Group the powers of 10 together and use the product rule to multiply them together

• Write in scientific notation

• Example: $5\times10^{16}\ times\ 3\times10^{-6}$  

•  $3\times10^5\ times\ 5\times10^{12}=\left(3\times5\right)\times\left(10^{5+12}\right)=15\times10^{17}$  

• Write in standard form...

•  $1.5\times10^{18}$  

Multiple Choice

There are about 130 million books in the world according to a 2010 study by Google. The average book has around 400 pages.

About how many pages are there in all the books in the world?

Select the correct answer that is expressed in SCIENTIFC NOTATION

•  $5.2\times10^{10}$  

•  $52\times10^9$  

•  $4\times10^2$  

•  $130\times10^6$