Exponential Algebra

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Exponential notation and algebra, including laws for zero and negative exponents, multiplying and dividing powers with the same base, products and quotients with the same exponent, powers of powers, equating exponents in exponential equations, and rationalizing denominators when needed.

Want a deeper conceptual understanding? Try our interactive lesson!

Exponential Notation

SL Core 1.5

Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:

an=a×a×⋯×an times🚫

Here a is called the base and n the exponent. We say that a is raised to the nth power.

Note that a1=a, since we have 1×a=a.

Exponent with zero

SL Core 1.5

Any number raised to the power zero is

a0=1×a×a×⋯×a0 times=1🚫

And since any number multiplied by 0 is 0:

0n=0,n=0🚫

When n=0, we have 00, which is technically undefined, but in most contexts is defined to be

00=1🚫

Multiplying powers with the same base

SL Core 1.5

When multiplying exponentials with the same base, the following rule applies:

an⋅am =a×a×⋯×an times×a×a×⋯×am times =a×a×⋯×an+m times =am+n🚫

Exponential of exponential

SL Core 1.5

An exponential can be the base of another exponential:

(am)n=a×⋯×am times×⋯×a×⋯×am timesn times=anm🚫

Negative exponents

SL Core 1.5

a−n=an1,a=0🚫

Dividing exponents with the same base

SL Core 1.5

In general,

aman=an⋅a−m=an−m,a=0🚫

Exponents of products & quotients

SL Core 1.5

When exponentials with the same power are being multiplied or divided, the bases can be combined:

anbn=(ab)n🚫

bnan=(ba)n,b=0🚫

Exponential Equations (Equating Indices)

SL AA 1.7

If two exponentials in the same positive base are equal, their exponents must be equal:

an=am⇔n=m,a>0,a=1🚫

Exponentials can also appear in equations with one or more unknown:

(21)x−1=8x+1

⇒(2−1)x−1=(23)x+1

⇒21−x=23x+3

Now we can equate the exponents:

1−x=3x+3⇒x=−21

Nice work completing Exponential Algebra, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Radicals and Roots

Exponents & Logarithms

Exponential Algebra

0 of 0 exercises completed

Want a deeper conceptual understanding? Try our interactive lesson!

Exponential Notation

SL Core 1.5

Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:

an=a×a×⋯×an times🚫

Here a is called the base and n the exponent. We say that a is raised to the nth power.

Note that a1=a, since we have 1×a=a.

Exponent with zero

SL Core 1.5

Any number raised to the power zero is

a0=1×a×a×⋯×a0 times=1🚫

And since any number multiplied by 0 is 0:

0n=0,n=0🚫

When n=0, we have 00, which is technically undefined, but in most contexts is defined to be

00=1🚫

Multiplying powers with the same base

SL Core 1.5

When multiplying exponentials with the same base, the following rule applies:

an⋅am =a×a×⋯×an times×a×a×⋯×am times =a×a×⋯×an+m times =am+n🚫

Exponential of exponential

SL Core 1.5

An exponential can be the base of another exponential:

(am)n=a×⋯×am times×⋯×a×⋯×am timesn times=anm🚫

Negative exponents

SL Core 1.5

a−n=an1,a=0🚫

Dividing exponents with the same base

SL Core 1.5

In general,

aman=an⋅a−m=an−m,a=0🚫

Exponents of products & quotients

SL Core 1.5

When exponentials with the same power are being multiplied or divided, the bases can be combined:

anbn=(ab)n🚫

bnan=(ba)n,b=0🚫

Exponential Equations (Equating Indices)

SL AA 1.7

If two exponentials in the same positive base are equal, their exponents must be equal:

an=am⇔n=m,a>0,a=1🚫

Exponentials can also appear in equations with one or more unknown:

(21)x−1=8x+1

⇒(2−1)x−1=(23)x+1

⇒21−x=23x+3

Now we can equate the exponents:

1−x=3x+3⇒x=−21

Nice work completing Exponential Algebra, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Radicals and Roots

Exponents & Logarithms

Exponential Algebra

0 of 0 exercises completed

Want a deeper conceptual understanding? Try our interactive lesson!

Exponential Notation

SL Core 1.5

Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:

an=a×a×⋯×an times🚫

Here a is called the base and n the exponent. We say that a is raised to the nth power.

Note that a1=a, since we have 1×a=a.

Exponent with zero

SL Core 1.5

Any number raised to the power zero is

a0=1×a×a×⋯×a0 times=1🚫

And since any number multiplied by 0 is 0:

0n=0,n=0🚫

When n=0, we have 00, which is technically undefined, but in most contexts is defined to be

00=1🚫

Multiplying powers with the same base

SL Core 1.5

When multiplying exponentials with the same base, the following rule applies:

an⋅am =a×a×⋯×an times×a×a×⋯×am times =a×a×⋯×an+m times =am+n🚫

Exponential of exponential

SL Core 1.5

An exponential can be the base of another exponential:

(am)n=a×⋯×am times×⋯×a×⋯×am timesn times=anm🚫

Negative exponents

SL Core 1.5

a−n=an1,a=0🚫

Dividing exponents with the same base

SL Core 1.5

In general,

aman=an⋅a−m=an−m,a=0🚫

Exponents of products & quotients

SL Core 1.5

When exponentials with the same power are being multiplied or divided, the bases can be combined:

anbn=(ab)n🚫

bnan=(ba)n,b=0🚫

Exponential Equations (Equating Indices)

SL AA 1.7

If two exponentials in the same positive base are equal, their exponents must be equal:

an=am⇔n=m,a>0,a=1🚫

Exponentials can also appear in equations with one or more unknown:

(21)x−1=8x+1

⇒(2−1)x−1=(23)x+1

⇒21−x=23x+3

Now we can equate the exponents:

1−x=3x+3⇒x=−21

Nice work completing Exponential Algebra, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

1 free

Radicals and Roots

Exponents & Logarithms

Exponential Algebra

0 of 0 exercises completed

Want a deeper conceptual understanding? Try our interactive lesson!

Exponential Notation

SL Core 1.5

Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:

an=a×a×⋯×an times🚫

Here a is called the base and n the exponent. We say that a is raised to the nth power.

Note that a1=a, since we have 1×a=a.

Exponent with zero

SL Core 1.5

Any number raised to the power zero is

a0=1×a×a×⋯×a0 times=1🚫

And since any number multiplied by 0 is 0:

0n=0,n=0🚫

When n=0, we have 00, which is technically undefined, but in most contexts is defined to be

00=1🚫

Multiplying powers with the same base

SL Core 1.5

When multiplying exponentials with the same base, the following rule applies:

an⋅am =a×a×⋯×an times×a×a×⋯×am times =a×a×⋯×an+m times =am+n🚫

Exponential of exponential

SL Core 1.5

An exponential can be the base of another exponential:

(am)n=a×⋯×am times×⋯×a×⋯×am timesn times=anm🚫

Negative exponents

SL Core 1.5

a−n=an1,a=0🚫

Dividing exponents with the same base

SL Core 1.5

In general,

aman=an⋅a−m=an−m,a=0🚫

Exponents of products & quotients

SL Core 1.5

When exponentials with the same power are being multiplied or divided, the bases can be combined:

anbn=(ab)n🚫

bnan=(ba)n,b=0🚫

Exponential Equations (Equating Indices)

SL AA 1.7

If two exponentials in the same positive base are equal, their exponents must be equal:

an=am⇔n=m,a>0,a=1🚫

Exponentials can also appear in equations with one or more unknown:

(21)x−1=8x+1

⇒(2−1)x−1=(23)x+1

⇒21−x=23x+3

Now we can equate the exponents:

1−x=3x+3⇒x=−21

Nice work completing Exponential Algebra, here's a quick recap of what we covered:

Skills covered

Mixed Practice

Exercises checked off

I'm Plex, here to help you understand this concept!

Generating starter questions...

1 free