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Concept — Integer Base
9 × 9 × 9 × 9 × 9
=
9
The exponent tells you how many times to write the base as a factor.
Concept — Fraction Base
13 × 13 × 13 × 13
=
(13)
=
13
=
3
The exponent still counts how many times you write the entire base — including when the base is a fraction.
Concept — Variable Base
Expanded Form
x * x * x * x * x
=
Exponential Form
x
This works for any base — numbers, fractions, or variables!
Vocabulary — Reading Exponents
b n base exponent (power)
"b to the power of n"  ·  "b to the nth"
4⁵
"four to the fifth power"
x³
"x cubed"
or "x to the third power"
(13)⁴
"one-third to the fourth power"
Type-In — Write 4² in Expanded Form

Write this in expanded form:

4²

Accepted: ×   *   or   x   between factors

Teacher / Amplify setup notes
Computation call: checkExpandedForm(studentInput, 4, 2)
Correct answers accepted: 4 × 4   4*4   4x4
Type-In — Write 2³ in Expanded Form

Write this in expanded form:

2³

Accepted: ×   *   or   x   between factors

Teacher / Amplify setup notes
Computation call: checkExpandedForm(studentInput, 2, 3)
Correct answers accepted: 2 × 2 × 2   2*2*2   2x2x2
Type-In — Write a × a × a in Exponent Form

Write this using an exponent:

a × a × a

Use ^ to write the exponent (e.g. x^2 means x²)

Teacher / Amplify setup notes
Computation call: checkExponentForm(studentInput, "a", 3)
Correct answers accepted: a^3   a³
Multiple Choice — (1/2)³ in Expanded Form

Which shows (1/2)³ written in expanded form?

(12)³
Teacher / Amplify setup notes
Computation call: problem_trans7(selectedValue)
Option values: 1 = "1/2 × 1/2 × 1/2" ✓  ·  2 = "1/6"  ·  3 = "3/2"  ·  4 = "1/8"
Multiple Choice — x * x * x * x * x in Exponent Form

Write this using an exponent:

x * x * x * x * x
Teacher / Amplify setup notes
Computation call: problem_trans8(selectedValue)
Option values: 1 = "x⁵" ✓  ·  2 = "5x"  ·  3 = "x + 5"  ·  4 = "5^x"
Multiple Choice — Repeated Fraction in Exponent Form

Which is the exponent form of this expression?

13 × 13 × 13 × 13
Teacher / Amplify setup notes
Computation call: problem_trans9(selectedValue)
Option values: 1 = "(1/3)⁴" ✓  ·  2 = "4/3"  ·  3 = "3⁴"  ·  4 = "1/81"
Type-In — Write Using a Negative Exponent

The expression below equals 3 raised to a negative power. Fill in the exponent:

13 × 13 × 13 × 13  =  3−

Enter the number that goes in the box (just the number — the − sign is already there)

Teacher / Amplify setup notes
Computation call: checkNegativeExp(studentInput, 4)
Answer: 4 — completes 3^(−4)
Visual — Teacher Examples
√9 = 3 × 3
9 3 × 3 √9 = 3 × 3, so √9 = 3
∛27 = 3 × 3 × 3
27 3 × 3 × 3 ∛27 = 3 × 3 × 3, so ∛27 = 3
⁴√16 = ? × ? × ? × ?
16 4 × 4 √16 = 4 ? × ? × ? × ? ⁴√16 = ? 16 = 4 × 4. What does each 4 split into?
Visual — Equivalent Notation
Roots and fractional exponents are two ways to write the same thing — they are always equal!
Example A — 27⅓ in every box
27 27⅓ × 27⅓ × 27⅓ fractional exp. fractional exp. fractional exp. 27⅓ × 27⅓ × 27⅓ = 27 ✓
Example B — Three names for the same number
8 ∛8 × 2 × 8⅓ radical notation value fractional exp. ∛8 = 2 = 8⅓ · all equal 2, and 2×2×2 = 8 ✓
Visual — Find the Root Factors
√36 = ?
36 ? × ? Each ? is the same number.
∛8 = ?
8 ? × ? × ? Each ? is the same number.
⁴√81 = ?
81 ? × ? × ? × ? Each ? is the same number.
Show answer key
Visual — Find the Power
5² = ?
? 5 × 5 Multiply the factors to fill in the top.
5² =
4³ = ?
? 4 × 4 × 4 Multiply the factors to fill in the top.
4³ =
Show answer key