Evaluate expressions using properties of exponents

Key Notes :

🌟 Evaluate Expressions Using Properties of Exponents

What is an Exponent?

👉 An exponent tells how many times a number (called the base) is multiplied by itself.

Example:

  • 24=2×2×2×2=16
⚙️ Basic Terms

Base: The number that is multiplied.

  • 👉 In 53, base = 5

Exponent (Power): The number of times the base is used as a factor.

  • 👉 In 53, exponent = 3

Expression: A mathematical phrase with numbers, variables, and exponents.

  • 👉 Example: 3x2.x3
💡 Properties of Exponents

1️⃣ Product of Powers Property

  • When multiplying with the same base, add the exponents.
  • 👉 am×an=am+n
  • 📘 Example: 23×24=23+4=27=128

2️⃣ Quotient of Powers Property

  • When dividing with the same base, subtract the exponents.
  • 👉 am÷an=am−n
  • 📘 Example: 56÷52=56−2=54=625

3️⃣ Power of a Power Property

  • When a power is raised to another power, multiply the exponents.
  • 👉 (am)n=am×n
  • 📘 Example: (32)4=38=6561

4️⃣ Power of a Product Property

  • Distribute the exponent to each factor inside the parentheses.
  • 👉 (ab)m=am×bm
  • 📘 Example: (2x)3=23×x3=8×3

5️⃣ Power of a Quotient Property

  • Apply the exponent to both numerator and denominator.
  • 👉 (a/b)m=am/bm
  • 📘 Example: (3/2)2=32/22=9/4

6️⃣ Zero Exponent Rule

  • Any nonzero number raised to the power of 0 is 1.
  • 👉 a0=1
  • 📘 Example: 90=1

7️⃣ Negative Exponent Rule

  • A negative exponent means take the reciprocal of the base.
  • 👉 a−m=1/am
  • 📘 Example: 2−3=1/23=1/8
🧠 Steps to Evaluate Exponential Expressions

1️⃣ Identify the base and exponent(s).

2️⃣ Apply the rules of exponents (add, subtract, multiply).

3️⃣ Simplify numerical values.

4️⃣ If variables are present, simplify their powers using the same base rule.

🎯 Example Problems
  1. 32×34=36=729
  2. (x3)2=x6
  3. y8/y3=y5
  4. (2a2b)3=23a6b3=8a6b3
🌈 Key Tips

✨ Always check if the bases are the same before applying rules.

✨ Simplify step by step to avoid mistakes.

✨ Remember: These rules work only when bases are identical!

Learn with an example

🔥Evaluate. Write your answer as a whole number or as a simplified fraction.

125 /123 =

To divide powers with the same base, subtract their exponents.

First, combine powers with the same bases.

 125 / 123= 125 − 3 —–> Divide the 12’s, remembering to subtract the exponents

  = 122

Now evaluate.

122 = 144

🔥Evaluate. Write your answer as a whole number or as a simplified fraction.

42 · 42 = 

To multiply powers with the same base, keep the base and add the exponents.

Evaluate.

42 · 42 = 16 · 42 ——->Evaluate 42

= 16 · 16 ———->Evaluate 42

= 256 ——>Multiply

🔥Evaluate. Write your answer as a whole number or as a simplified fraction.

27/24=

First combine powers with the same bases.

27 / 24 = 27-4 —–> Divide the 2’s, remembering to subtract the exponents

= 23

Now evaluate.

23 = 8

Let’s practice!