Short Cuts with Exponents

Shortcuts with Exponents

The goal is to develop algebraic intuition and understanding rather than just trying to get the answer as fast as possible.

Multiple Choice

How would you write $$6\times6\times6\times6\times6\times6\times6\times6$$ using an exponent?

• 8⁶

• 6⁸

• 48

• 1,679,616

• 36

Multiple Choice

The base number in 3⁵ is

• 3

• 5

• 243

​Watch or Read the Example

Multiple Choice

Simplify using the rules of exponents

$$\frac{2^3\times2^7}{2^6\times2^3}=$$

• $$\frac{1}{2}$$

• $$1$$

• $$2$$

• $$2^2$$

Multiple Choice

Simplify using the rules of exponents

$$\frac{2^3\times2^7}{2^6\times2^3}=$$

• $$2$$

• $$2^2$$

Multiple Choice

Which of these numbers is larger?

• $$3^4+3^4+3^4$$

• $$3^5$$

• These numbers are equal

Multiple Choice

What is the value of

$$2^{8^2}$$

• $$2^{16}$$

• $$2^{24}$$

• $$2^{64}$$

• $$2^{72}$$

Multiple Choice

If

$$\frac{2^z}{2^y\times2^z}=2$$

which of these math sentences are true?

• $$x-y-z=0$$

• $$x-y-z=1$$

• $$x-y+z=0$$

• $$x-y+z=1$$

Multiple Select

$$5^{-7}\times5^7=?$$

• $$0$$

• $$1$$

• $$5^2$$

• $$5^{-49}$$

• $$5^0$$

Multiple Choice

Which is smaller?

• $$2^{-3}$$

• $$3^{-2}$$

• They are the same.

Poll

$$12-2=10$$

$$12+\left(-2\right)=10$$

Adding the opposite is the same as subtracting.

Multiple Choice

$$4^3\times4^{-3}=$$

• $$-1$$

• $$0$$

• $$1$$

• $$64$$

Multiple Choice

$$64\times4^{-3}=1$$

What is the value of $$4^{-3}$$ ?

• $$-12$$

• $$\frac{1}{64}$$

• $$\frac{4}{3}$$

• $$64$$

Multiple Select

$$5\times2^3\times2^{-2}=$$

There is more than one correct answer.

• $$\frac{5\times8}{4}$$

• $$\frac{5\times2^3}{2^2}$$

• $$\frac{2^2}{5\times2^3}$$

• $$10$$

Multiple Select

$$5\times2^3\times2^{-2}=$$

• $$\frac{5\times8}{4}$$

• $$\frac{5\times2^3}{2^2}$$

• $$10$$

• $$10^0$$

Multiple Choice

What is the value of

$$\left(\frac{2}{3}\right)^{-1}$$

• $$-\frac{3}{2}$$

• $$-\frac{2}{3}$$

• $$\frac{-2}{-3}$$

• $$\frac{3}{2}$$

A negative sign in the exponent tells you that you are looking for a reciprocal.

The number part of the exponent tells you how many factors you have.

Be careful.

​Here's Why.

Match

Match the following

• $$\left(\frac{1}{3}\right)^{-1}$$

• $$\left(\frac{3}{4}\right)^{-2}$$

• $$\left(\frac{1}{3}\right)^{-2}$$

• $$\left(\frac{3}{4}\right)^{-1}$$

• $$\left(\frac{1}{3}\right)^{-1}\left(\frac{3}{4}\right)^{-1}$$

3

9/16

9

4/3

4

• 3

• 9/16

• 9

• 4/3

• 4

Multiple Choice

Which is Larger?

$$5^{-2}\ or\ \ 2^{-5}$$

• $$5^{-2}$$

• $$2^{-5}$$

• They are the same

Remember the fraction bar means DIVISION.

sO

The more you divide something up, the smaller each part is

​

Think of the size.

​Here's WHY

Multiple Choice

If $$\left(x^y\right)^3=x$$

what is the value of $$y$$ ?

• $$-2$$

• $$0$$

• $$\frac{1}{3}$$

• $$1$$

• none of these are correct

You need three equal factors to make a perfect cube.

​

one of three equal whatevers

​Here's why.

Multiple Choice

Which value is the largest?

• $$\left(-1\right)^{99}$$

• $$\left(-1\right)^8$$

• $$\left(\frac{1}{2}\right)^3$$

• $$0$$

• They are all the same.

What do you notice in the pattern of powers of a negative base.

See how the negative sign alternates with the positive sign. Why do you suppose that is?

Here's why.

Multiple Choice

$$\frac{\left(-3\right)^4\times\left(-3\right)^{\frac{6}{5}}}{\left(-3\right)^{\frac{1}{5}}\times\left(-3\right)^3}=$$

• $$-9$$

• $$-3$$

• $$3$$

• $$9$$

Rewrite the division problem, so the exponents are easy to subtract.

​

​Some text here about the topic of discussion.

Here's why.

Multiple Choice

True or False?

$$4^{\frac{2}{3}}=\left(\sqrt[3]{4}\right)^2$$

• True

• False

• You have got to be kidding

Numbers come in different forms that mean the same thing:

​in this case, exponential form and radical form.

​Some text here about the topic of discussion.

Here's why.

Poll

No matter what real number is used, if you square it, the square is ALWAYS bigger.

$$m^2>m\ \ \ \ \ ALWAYS!$$

• Truth

• Lie

• Misunderstanding

• Sometimes

​m can be any real number:

positive, negative, a fraction, so Sometimes is the best answer.

half of a half is a quarter

​again think about size of the pieces

​Here's why.

Multiple Choice

$$\frac{3^{-1}\times5^{-\frac{1}{2}}\times9^1}{3^1\times5^{\frac{1}{2}}\times9^{-1}}$$

Evaluate

• $$\frac{5}{9}$$

• $$\frac{3}{5}$$

• $$\frac{9}{5}$$

• $$\frac{5}{3}$$

Use the exponent law for division and subtract the exponents of like bases.

-2 is 2 factors of 3 in the denominator (because it is negative)

-1 is 1 factor of 5 in the denominator

2 is 2 factors of 9 in the numerator (because it is positive)

​

integers can be tricky be careful

Here's why.

Multiple Choice

Given the expression below, which action will produce a larger value?

$$3^3\times5^2$$

• Doubling both bases

• Doubling both exponents

• Neither will change

Exponent towers are expressions in which a base is raised to a power, which is raised to one or more powers.

​

Recall 2⁰ is equal to 1.

​Powers & Towers

Multiple Choice

Which has the greatest value?

• $$2^{3^4}$$

• $$7^{8^{-2}}$$

• $$\left(-5\right)^{\left(\frac{1}{2}\right)^4}$$

• $$2^{3^{4^{5^6}}}$$

Multiple Choice

Which has the greatest value?

• $$2^{\left(3^4\right)}$$

• $$\left(2^3\right)^4$$

• They are the same value.

Multiple Choice

$$2^{4^{\left(\frac{1}{2}\right)^{5^0}}}$$

• $$1$$

• $$2$$

• $$4$$

• $$5$$

Multiple Choice

which is greater?

$$3^{2^3}$$ or $$3^{3^2}$$ ?

• $$3^{2^3}$$

• $$3^{3^2}$$

• They are equivalent.

Multiple Choice

$$m=\left(\frac{1}{2}\right)^{\left(\frac{1}{2}\right)^{2^2}}$$

• $$m>1$$

• $$m=1$$

• $$-1<m<1$$

• $$m=-1$$

• $$m<-1$$

Multiple Choice

$$\left(-1\right)^{\left(-1\right)^{\left(-1\right)}}\ \ \$$ □ $$\left(-1\right)^{\left(-1\right)^{\left(-1\right)^{\left(-1\right)^{\left(-1\right)}}}}$$

What symbol belongs in the box?

• $$<$$

• $$=$$

• $$>$$

Multiple Choice

$$\left(-3\right)^{-1^{-3^{-1}}}=$$

• $$-3$$

• $$-1$$

• $$-\frac{1}{3}$$

• $$\frac{1}{3}$$

• $$1$$

Multiple Choice

$$4^x+5^x+6^x=3$$

What is the value of x?

• $$-3$$

• $$-2$$

• $$-1$$

• $$0$$

Multiple Choice

$$\sqrt[]{9^{36x^4}}=$$ ?

• $$3^{6x^2}$$

• $$9^{6x^2}$$

• $$9^{18x^2}$$

• $$9^{18x^4}$$

Multiple Choice

$$2^x<\left(\frac{1}{2}\right)^x$$

What values of x make this a true statement?

• $$x<0$$

• $$x>0$$

• There are no solutions.

Multiple Choice

If $$2^x=10$$ and $$10^y=32$$ , what is the value of $$xy$$ ?

• $$4$$

• $$5$$

• $$8$$

• $$10$$

It is all about substitution.

We are changing the way it looks without changing what it means.

​

​change the form not the meaning

​Here's why.

Poll

How do you feel about the information in this Quizizz

Poll

The hardest to figure out was​ ​

• negative exponents

• towers of powers

• figuring out which was bigger

• fractions as exponents

• finding values