Division, Zero, And Negative Exponents

Understand how subtracting exponents corresponds to division, leading to zero and negative exponents.

8 lessons

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Concept

Erasing Half the Line

Introduce division of powers.

Imagine a straight line of length 16 units. Using exponential notation, we can write 16 units as $2^4$ units.

Division by Halving

If we erase half of it, we divide by 2. $2^4 \div 2 = \frac{2 \times 2 \times 2 \times 2}{2} = 2^3 = 8 \text{ units}$

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IMAGE

Division, Zero, and Negative Exponents Visual

The 'Power Lines' diagram is essential here to show how stepping left on the number line divides the value by the base and reduces the exponent by 1.

bold editorial infographic, clean data-forward design, high-contrast color blocks, elegant typography hierarchy, geometr…

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Law 4: Division of Powers

Formula for n^a / n^b.

n^{a} \div n^{b} = n^{a - b}

$n$ is the base ( $n \neq = 0$ )
$a, b$ are counting numbers

Extension:

m^{a} \div n^{a} = (\frac{m}{n})^{a}

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Concept

When Zero is in Power

Deriving the zero exponent.

We know what happens when an exponent is a positive counting number. But what is the value of $2^0$ ? Let's figure it out using the division law.

Proving the Zero Power

Let's divide $2^4$ by itself. According to our division rule, we subtract the exponents: $2^4 \div 2^4 = 2^{4-4} = 2^0$

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Example

The Negative Exponent

Deriving negative exponents.

Problem. What happens if we divide a smaller power by a larger one? Let's take a line of $2^4$ units (16 units) and halve it 5 times. That is $2^4 \div 2^5$ .

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reading

Simplify with Negatives

Faded example for simplifying expressions.

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To simplify the expression ( 3^2 \times 3^{-5} \times 3^6 ), we can use the product rule for exponents. This rule states that when multiplying terms with the same base, we add their exponents together, such that ( n^a \times n^b = n^{a+b} ). First, we set up the addition of our exponents: ( 2 + (-5) + 6 ). Calculating this sum gives us the new exponent, which is . Now, we apply this exponent to our base of 3 to get the simplified exponential form. Finally, if we need to evaluate this expression completely, we calculate 3 multiplied by itself three times. This yields a final numerical value of .

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quiz

True or False?

Conceptual check on exponents.

1 / 5

Identify the greater number: $100^2$ or $2^{100}$ .

← previousMultiplication Laws Of Exponents next →Scientific Notation And Powers Of 10

Concept

Erasing Half the Line

Introduce division of powers.

Imagine a straight line of length 16 units. Using exponential notation, we can write 16 units as $2^4$ units.

Division by Halving

If we erase half of it, we divide by 2. $2^4 \div 2 = \frac{2 \times 2 \times 2 \times 2}{2} = 2^3 = 8 \text{ units}$

1 / 3

Concept

When Zero is in Power

Deriving the zero exponent.

We know what happens when an exponent is a positive counting number. But what is the value of $2^0$ ? Let's figure it out using the division law.

Proving the Zero Power

Let's divide $2^4$ by itself. According to our division rule, we subtract the exponents: $2^4 \div 2^4 = 2^{4-4} = 2^0$

1 / 3

Example

The Negative Exponent

Deriving negative exponents.

Problem. What happens if we divide a smaller power by a larger one? Let's take a line of $2^4$ units (16 units) and halve it 5 times. That is $2^4 \div 2^5$ .

1 / 5

quiz

True or False?

Conceptual check on exponents.

1 / 5

Identify the greater number: $100^2$ or $2^{100}$ .