Simplify the Expression: 3^(2a) ÷ 3^a Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it as canceling out repeated multiplication! When you divide $ 3^4 ÷ 3^2 $, you're canceling two 3's from the bottom with two 3's from the top, leaving $ 3^2 $.

Q: What if the variable expressions are more complex?

The rule still works! For $ \frac{x^{3a+2}}{x^{a-1}} $, you'd get $ x^{(3a+2)-(a-1)} = x^{2a+3} $. Just be careful with parentheses when subtracting!

Q: Can I use this rule backwards?

Absolutely! If you see $ 5^{3x} $, you could write it as $ \frac{5^{5x}}{5^{2x}} $ or many other combinations. This helps with factoring and solving equations.

Q: What happens if the exponents are equal?

Great question! If you have $ \frac{3^a}{3^a} $, you get $ 3^{a-a} = 3^0 = 1 $. Any non-zero number to the power of 0 equals 1!

Q: How can I check my answer is right?

Pick a simple value for the variable! If a=1, then $ \frac{3^2}{3^1} = \frac{9}{3} = 3 $, and $ 3^1 = 3 $. They match! Always test with easy numbers first.

Exponent Division with Variable Expressions

Insert the corresponding expression:

$\frac{3^{2a}}{3^a}=$

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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)

00:09 We'll use this formula in our exercise

00:11 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:

$\frac{3^{2a}}{3^a}=$

Step-by-step solution

To solve the question, let's apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases with exponents, you can subtract the exponent of the denominator from the exponent of the numerator. Mathematically, this is expressed as:

$\frac{b^m}{b^n} = b^{m-n}$

In our case, the base $b$ is 3, the exponent $m$ for the numerator is $2a$ , and the exponent $n$ for the denominator is $a$ . Thus, we can substitute these into the formula:

$\frac{3^{2a}}{3^a} = 3^{2a-a}$

Now, simplify the exponent:

$3^{2a-a} = 3^{a}$

Therefore, the expression simplifies to:

$3^a$

The solution to the question is: $3^a$

Final Answer

$3^a$

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: Subtract denominators from numerators: $3^{2a-a} = 3^a$
Check: Test with specific values like a=2: $\frac{3^4}{3^2} = \frac{81}{9} = 9 = 3^2$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of subtracting exponents
Don't add the exponents (2a + a = 3a) when dividing = $3^{3a}$ ! Division requires subtraction, not addition. 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 as canceling out repeated multiplication! When you divide $3^4 ÷ 3^2$ , you're canceling two 3's from the bottom with two 3's from the top, leaving $3^2$ .

What if the variable expressions are more complex?

The rule still works! For $\frac{x^{3a+2}}{x^{a-1}}$ , you'd get $x^{(3a+2)-(a-1)} = x^{2a+3}$ . Just be careful with parentheses when subtracting!

Can I use this rule backwards?

Absolutely! If you see $5^{3x}$ , you could write it as $\frac{5^{5x}}{5^{2x}}$ or many other combinations. This helps with factoring and solving equations.

What happens if the exponents are equal?

Great question! If you have $\frac{3^a}{3^a}$ , you get $3^{a-a} = 3^0 = 1$ . Any non-zero number to the power of 0 equals 1!

How can I check my answer is right?

Pick a simple value for the variable! If a=1, then $\frac{3^2}{3^1} = \frac{9}{3} = 3$ , and $3^1 = 3$ . They match! Always test with easy numbers first.

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