Simplify the Power Expression: 2²/2⁶ Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{2^2}{2^6} = \frac{2 \times 2}{2 \times 2 \times 2 \times 2 \times 2 \times 2} $. The two 2's on top cancel with two 2's on bottom, leaving four 2's on bottom = $ \frac{1}{2^4} $!

Q: What does a negative exponent mean?

A negative exponent means "flip it to the denominator"! So $ 2^{-4} = \frac{1}{2^4} $. It's like the number is asking to move to the bottom of a fraction.

Q: Can I just cancel the 2's directly?

Not recommended! While you could cancel, using the quotient rule $ \frac{a^m}{a^n} = a^{m-n} $ is more reliable and works for all exponent problems, even complex ones.

Q: How do I remember which exponent goes first in subtraction?

Remember: "Top minus Bottom"! The numerator exponent (top) minus the denominator exponent (bottom). So $ \frac{2^2}{2^6} $ becomes $ 2^{2-6} = 2^{-4} $.

Q: What if both exponents are the same?

Great question! If exponents are equal, like $ \frac{2^3}{2^3} $, you get $ 2^{3-3} = 2^0 = 1 $. Any number to the zero power equals 1!

Exponent Division with Negative Powers

Insert the corresponding expression:

$\frac{2^2}{2^6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 First, let's simplify.

00:10 We'll use the formula for dividing powers.

00:14 If you have A to the power of N, divided by A to the power of M...

00:19 It equals A to the power of M minus N.

00:23 Let's apply this to our exercise.

00:26 Now, the formula for negative exponents.

00:29 A to the power of negative N means...

00:33 You use the reciprocal, one over A, to the power of N.

00:38 Apply this formula in our exercise.

00:42 Substitute the reciprocal and the positive power.

00:45 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{2^2}{2^6}=$

Step-by-step solution

Let's solve the expression $\frac{2^2}{2^6}$ using the rules of exponents. Specifically, we'll use the Power of a Quotient Rule for Exponents which states that $\frac{a^m}{a^n} = a^{m-n}$ .

First, identify the base, which is 2, and the exponents. According to the rule, we subtract the exponent in the denominator from the exponent in the numerator.
In our case, the exponents are 2 (in the numerator) and 6 (in the denominator).
Subtract the exponent in the denominator from the exponent in the numerator: $2 - 6 = -4$ . This gives us $2^{-4}$ .
According to the rule of negative exponents, $a^{-n} = \frac{1}{a^n}$ , so we rewrite $2^{-4}$ as $\frac{1}{2^4}$ .

Therefore, the expression $\frac{2^2}{2^6}$ simplifies to $\frac{1}{2^4}$ .

The solution to the question is: $\frac{1}{2^4}$

Final Answer

$\frac{1}{2^4}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Calculate exponent first: $2 - 6 = -4$ , then apply negative exponent rule
Check: Verify $\frac{1}{2^4} = \frac{1}{16}$ equals $\frac{4}{64} = \frac{1}{16}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 2 + 6 = 8 to get $2^8$ ! This gives 256 instead of 1/16. Adding is for multiplication, not division. Always subtract the bottom exponent from the top exponent when dividing.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{2^2}{2^6} = \frac{2 \times 2}{2 \times 2 \times 2 \times 2 \times 2 \times 2}$ . The two 2's on top cancel with two 2's on bottom, leaving four 2's on bottom = $\frac{1}{2^4}$ !

What does a negative exponent mean?

A negative exponent means "flip it to the denominator"! So $2^{-4} = \frac{1}{2^4}$ . It's like the number is asking to move to the bottom of a fraction.

Can I just cancel the 2's directly?

Not recommended! While you could cancel, using the quotient rule $\frac{a^m}{a^n} = a^{m-n}$ is more reliable and works for all exponent problems, even complex ones.

How do I remember which exponent goes first in subtraction?

Remember: "Top minus Bottom"! The numerator exponent (top) minus the denominator exponent (bottom). So $\frac{2^2}{2^6}$ becomes $2^{2-6} = 2^{-4}$ .

What if both exponents are the same?

Great question! If exponents are equal, like $\frac{2^3}{2^3}$ , you get $2^{3-3} = 2^0 = 1$ . Any number to the zero power equals 1!

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