Identify equivalent expressions involving exponents

Key Notes :

🌟 Identify Equivalent Expressions Involving Exponents

🔹 What are Equivalent Exponential Expressions?

👉 Two expressions are equivalent if they have the same value, even if they look different.

✅ Example:

  • 23×22=23+2=25
  • So, 23×22 and 25 are equivalent expressions.
🔹 Basic Exponent Rules (Laws of Exponents)

These rules help us find and simplify equivalent exponential expressions:

Rule NameFormulaExample
Product Ruleam×an=am+n32×34=36
Quotient Ruleam/an=am−n56/52=54
Power Rule(am)n=am×n(23)2=26
Zero Exponent Rulea0=1(where a≠0)70=1
Negative Exponent Rulea−n=1/an4−2=1/42=1/16
Power of a Product(ab)m=am×bm(2×3)2=22×32
Power of a Quotient(a/b)m=am/bm(2/5)3=23/53
🔹 How to Identify Equivalent Expressions

👉 To check if two exponential expressions are equivalent, use these steps:

  1. Simplify both expressions using exponent rules.
  2. Compare the base and the exponent.
  3. If both are the same after simplification → ✅ Equivalent!

🧮 Example:

  • Check if (x2)3 and x6 are equivalent.
  • Simplify: (x2)3=x2×3=x6
  • ✅ Both are equivalent.
🔹 Common Mistakes to Avoid ❌

⚠️ Don’t add or multiply exponents when bases are different.

  • Example: 23+33≠53 ❌

⚠️ Remember that (a+b)n≠an+bn

  • Example: (2+3)2=25, but 22+32=13 — not equal!
🔹 Real-Life Connection 🌍

Exponents are used to express:

  • Area and volume formulas (e.g., s2, s3)
  • Scientific notation (3×105)
  • Growth patterns (population, bacteria growth, etc.)
✨ Summary
  • Equivalent expressions have the same value.
  • Use exponent rules to simplify expressions.
  • Always check bases and exponents.
  • Avoid common exponent errors.
🧠 Example Practice
  1. Are 23×24 and 27 equivalent? ✅
  2. Is (x2)3=x6? ✅
  3. Is 34÷32=32? ✅
  4. Is (2×3)2=22+32? ❌

Learn with an example

Select all the expressions that are equivalent to 85,3-3

  • 1/8-9
  • 8-9
  • 8
  • 80

To find which expressions are equivalent to 83.8-3, write each expression with the same base.

The answer choices and 83 . 8 –3 can all be written as a power of 8. So, use properties of exponents to write each expression in the form 8 ( ). Then, see which ones match 83 . 8 –3 in this form.

Step 1: Write 83.8–3 in the form 8( )

83.8–3

= 83+-3 Use the identity axay = ax+y

= 80

A positive number raised to zeroth power is 1. So, 83 . 8-3  is equal to 1.

Step 2: Now see which expressions are also equal to 80 = 1.

See if 1 / 8-9 is equal to 1.

1 / 8-9

=8-(-9) Use the identity 1/ax=a–x

= 89

So, 1 / 8-9 is equal to 89, but 83 . 8-3 is not. (It is equal to 80=1.)
The expressions1 / 8-9and 83 . 8-3are not equivalent.

The expression 8-9 is in the form 8 . You’ve already found that 83 . 8-3 is not equal to 8-9 (It is equal to 80=1)

The expression 8-9 and  83 . 8-3 are not equivalent.

8 is not equal to 1 , but 83 . 8-3 is equal to 1

The expression 8 and 83 . 8-3 are not equivalent.

you’ve already found 83 . 8-3 is equal to 80 .

The expression 80 and  83 . 8-3 are equivalent.

The correct answer is 80.

Select all the expressions that are equivalent to (103)0.

  • 10
  • 1/100
  • 1
  • 1/103

To find which expressions are equivalent to (103)0, write each expression with the same base.

The answer choices and (103)0 can all be written as a power of 10. So, use properties of exponents to write each expression in the form 10 ( )
. Then, see which ones match (103)0 in this form.

Step 1: Write (103)0 in the form 10.

(103)0

= 103.0 Use the identity (ax)y=axy

=100

A positive number raised to zero power is 1. So, (103)0 is equal to 1.

Step 2: Now see which expressions are also equal to 100=1.

10 is not equal to 1, but (103)0 is equal to 1.

The expressions 10 and (103)0 are not equivalent.

See if 1/100 is equal to 1.

1/100

1/1 Use the identity a0=1

1

Both 1/100 and (103)0 are equal to 1.

The expressions 1/100 and (103)0 are equivalent.

You’ve already found that (103)0 is equal to 1.
The expressions 1 and (103)0 are equivalent.

See if 1/103 is equal to 1.

1/103

=10–3

Use the identity
1/ax=a–x

So, 1/103 is equal to 10–3, but (103)0 is not. (It is equal to 100=1.)
The expressions 1/103 and (103)0 are not equivalent.

The correct answers are:

=1/100

=1

Let’s practice!