Unit 2 · NumbersN3 · Exponents & Radicals · Propulsion

Exponents & Scientific Notation

Apply exponent laws, compare magnitudes, and keep scientific notation normalized.

29 min
Unlocks: Turbopumps
Desmos way

Ready when you are

7 steps · about 29 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

Finish the previous step to continue, or use “Show everything” above to explore.

Key terms

Base: The repeated factor in an exponential expression.
Exponent: The number indicating how many equal factors are used.
Scientific notation: A number written as a coefficient from 1 to 10 times a power of 10.

Step 2 of 7Try it firstDone

Try it first

Finish the previous step to continue, or use “Show everything” above to explore.

Take a shot before the lesson. Write one relationship from the stem, even if you do not finish.

Step 3 of 7LearnDone

The big idea

Finish the previous step to continue, or use “Show everything” above to explore.

See the structure

For the same base, multiplication adds exponents and division subtracts them. A power of a power multiplies exponents.

Key idea

Name quantities before calculating

Write what each number measures. A correct calculation with the wrong quantity or unit still misses the question.

Build a reliable method

A negative exponent means reciprocal; it does not make the value negative. A zero exponent is 1 when the base is nonzero.

Explore in Desmos

Verify exponent structure

All three lines display 81, but for different valid reasons. The first two show that equal bases multiply by adding exponents; the third shows that squaring a product squares each factor.

The calculator is live: change anything and see what happens.

Verify exponent structure

Degree mode

Loading Desmos…

Stretch to a 36

Scientific notation has one nonzero digit before the decimal. Multiply coefficients and add powers of 10, then normalize.

Trap

A believable answer to the wrong question

The choices often include a correct intermediate number. Before selecting it, reread the highlighted ask and check the unit, sign, and domain.

Step 4 of 7Worked examplesDone

Worked examples

Finish the previous step to continue, or use “Show everything” above to explore.

Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Distinguish product from power rules

Problem

Which expression is equivalent to

x^3x^4

The shared base means add exponents:

x^{3+4}

Which exponent rule applies to this structure?

The product is

x^7

Which expanded or decimal form confirms the exponent result?

Worked example

Example 2: Normalize scientific notation after multiplying

Problem

What is the value of

(3\times10^5)(4\times10^{-2})

in normalized scientific notation?

Multiply coefficients and add exponents:

12\times10^3

Which exponent rule applies to this structure?

Normalize:

12\times10^3=1.2\times10^4

Which expanded or decimal form confirms the exponent result?

Faded example · you finish it

Example 3: Complete the setup and result

Problem

Which value is equivalent to

\dfrac{(2^3)^2}{2^4}

Choose the setup that follows the stated relationship.

Choose the resulting value.

Desmos way

Compare correct and incorrect exponent rules

Type (2^3)^2/2^4 and 2^(3*2-4). Both display 4 because a power of a power multiplies exponents and a quotient subtracts them.

Type 2^(3+2-4) and read 2. That mismatch exposes the common error of adding 3 and 2 inside the power of a power.

Compare correct and incorrect exponent rules

Degree mode

Loading Desmos…

Step 5 of 7PracticeDone

Practice

Finish the previous step to continue, or use “Show everything” above to explore.

Step 6 of 7Spot the mistakeDone

Spot the mistake

Finish the previous step to continue, or use “Show everything” above to explore.

Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

Finish the previous step to continue, or use “Show everything” above to explore.

Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.