Rules of exponents in math operations

Rules of exponents explained for 6th and 7th graders

Exponents are a way to show that a number is multiplied by itself several times. Instead of writing out the same number again and again, we use exponents to make it easier. For example, instead of writing 2 × 2 × 2, we can write 2³.

Here are the key rules of exponents you need to know, explained step by step:

1. The Product Rule (Multiplying with the Same Base)
When multiplying two numbers with the same base, add the exponents.

Rule:
aᵐ × aⁿ = aᵐ⁺ⁿ

• Base: The number that is being multiplied.
• Exponent: The small number that tells how many times the base is multiplied by itself.

Example:
2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

2. The Quotient Rule (Dividing with the Same Base)
When dividing two numbers with the same base, subtract the exponents.

Rule:
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (as long as m > n)

Example:
5⁶ ÷ 5² = 5⁶⁻² = 5⁴ = 625

3. The Power of a Power Rule
When raising a power to another power, multiply the exponents.

Rule:
(aᵐ)ⁿ = aᵐ × ⁿ

Example:
(3²)⁴ = 3² × ⁴ = 3⁸ = 6,561

4. The Power of a Product Rule
When you raise a product to a power, raise each factor in the product to that power.

Rule:
(ab)ᵐ = aᵐ × bᵐ

Example:
(2 × 3)⁴ = 2⁴ × 3⁴ = 16 × 81 = 1,296

5. The Power of a Quotient Rule
When raising a fraction to a power, raise both the numerator and the denominator to the power.

Rule:
(a/b)ᵐ = aᵐ / bᵐ

Example:
(3/4)² = 3² / 4² = 9/16

6. The Zero Exponent Rule
Any number raised to the power of zero is always 1 (as long as the base is not zero).

Rule:
a⁰ = 1

Example:
7⁰ = 1

This rule works for any number except zero, because 0⁰ is undefined.

7. The Negative Exponent Rule
A negative exponent means you take the reciprocal (flip the fraction) of the base and change the exponent to positive.

Rule:
a⁻ᵐ = 1/aᵐ

Example:
2⁻³ = 1/2³ = 1/8

8. The Identity Exponent Rule
Any number raised to the power of 1 is just the number itself.

Rule:
a¹ = a

Example:
9¹ = 9

Summary of Rules:

• Product Rule: Add the exponents when multiplying.
• Quotient Rule: Subtract the exponents when dividing.
• Power of a Power: Multiply the exponents.
• Power of a Product: Distribute the exponent to all factors.
• Power of a Quotient: Apply the exponent to both numerator and denominator.
• Zero Exponent: Any base to the power of zero equals 1.
• Negative Exponent: Flip the base and make the exponent positive.
• Identity Exponent: Any number raised to the power of 1 is itself.

These rules help simplify expressions with exponents and make it easier to calculate large powers. With these examples and rules, you can solve any exponent problem!