Simplify the Expression: 2^a ÷ 2^2 Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{2^5}{2^3} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2} $. The three 2's cancel out, leaving $ 2^2 $. That's 5 - 3 = 2!

Q: What if the variable is in the denominator exponent?

The rule still works! For example, $ \frac{2^3}{2^a} = 2^{3-a} $. Just remember: numerator exponent minus denominator exponent.

Q: Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. You can't simplify $ \frac{2^a}{3^2} $ using this rule.

Q: How do I remember which exponent to subtract from which?

Use the pattern: $ \frac{\text{top}}{\text{bottom}} = \text{base}^{\text{top} - \text{bottom}} $. Always subtract the bottom exponent from the top exponent.

Quotient Rule with Exponential Expressions

Insert the corresponding expression:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 We'll use this formula in our exercise

00:10 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

To solve the expression $\frac{2^a}{2^2}$ , we will apply the Power of a Quotient Rule for Exponents, which states that when you divide two powers with the same base, you subtract the exponents.

Here are the steps:

Identify the base: Both the numerator and the denominator have the same base, which is 2.
Apply the quotient rule of exponents: $\frac{b^m}{b^n} = b^{m-n}$ . By applying this rule:

$\frac{2^a}{2^2} = 2^{a-2}$ .

Thus, by utilizing the rule, we find that:

$\frac{2^a}{2^2} = 2^{a-2}$ .

The solution to the question is: $2^{a-2}$

Final Answer

$2^{a-2}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents: $\frac{b^m}{b^n} = b^{m-n}$
Technique: For $\frac{2^a}{2^2}$ , subtract exponents: a - 2 = $2^{a-2}$
Check: Verify by expanding: $2^{a-2} = \frac{2^a}{2^2}$ when multiplied by $2^2$ ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents like a + 2 = $2^{a+2}$ or multiply like 2a = $2^{2a}$ ! This confuses division rules with multiplication rules and gives completely wrong answers. Always remember: division means subtract the exponents when bases are the same.

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: $\frac{2^5}{2^3} = \frac{2 \times 2 \times 2 \times 2 \times 2}{2 \times 2 \times 2}$ . The three 2's cancel out, leaving $2^2$ . That's 5 - 3 = 2!

What if the variable is in the denominator exponent?

The rule still works! For example, $\frac{2^3}{2^a} = 2^{3-a}$ . Just remember: numerator exponent minus denominator exponent.

Can I use this rule with different bases?

No! The quotient rule only works when the bases are exactly the same. You can't simplify $\frac{2^a}{3^2}$ using this rule.

What happens if I get a negative exponent?

That's fine! A negative exponent means the reciprocal. For example, if a = 1, then $2^{1-2} = 2^{-1} = \frac{1}{2}$ .

How do I remember which exponent to subtract from which?

Use the pattern: $\frac{\text{top}}{\text{bottom}} = \text{base}^{\text{top} - \text{bottom}}$ . Always subtract the bottom exponent from the top exponent.

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