Negative Exponents Pattern with Powers of 5

✏️ Worked Example - Negative Exponents

Conceptual Pattern with Base 5
Notice how each step down in the exponent makes the value 5 times smaller, and each step up makes the value 5 times larger.

Step 1: Start with known positive powers.
\( 5^4 = 625,\; 5^3 = 125,\; 5^2 = 25,\; 5^1 = 5 \)

Step 2: Decrease the exponent by 1 each time.
Each step means divide by 5.
\( 625 \div 5 = 125 \)
\( 125 \div 5 = 25 \)
\( 25 \div 5 = 5 \)
\( 5 \div 5 = 1 \)

Step 3: Continue the same pattern past zero.
\( 1 \div 5 = \frac{1}{5} \)

Step 4: Keep dividing by 5 to get more negative exponents.
\( \frac{1}{5} \div 5 = \frac{1}{25} \)
\( \frac{1}{25} \div 5 = \frac{1}{125} \)

Conclusion:
Negative exponents do not mean the base is negative. They mean the value is the reciprocal of the corresponding positive power.

Key Pattern:
\( 5^{-1}=\frac{1}{5},\quad 5^{-2}=\frac{1}{25},\quad 5^{-3}=\frac{1}{125} \)

🧠 Math Vocabulary

• Base: The repeated factor in an exponential expression.
• Exponent: The small raised number that tells how many times the base is used as a factor.
• Power: The value of an exponential expression, such as \(5^3\).
• Reciprocal: Two numbers whose product is 1, such as \(5\) and \( \frac{1}{5} \).
• Negative Exponent: An exponent that indicates repeated division by the base and results in a reciprocal value.
• Zero Exponent: Any nonzero base raised to the zero power equals 1.

📚 What You Should Already Know

Students should already know how positive exponents work and understand that multiplication makes values larger while division makes values smaller.

🚀 What Comes Next

Next, students can apply this same pattern to simplify expressions with negative exponents and rewrite answers without negative powers.

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