Simplify the Expression: a⁸/a Using Exponent Rules

Q: Why do we subtract exponents when dividing?

When you divide powers, you're essentially canceling out common factors. Think of $ \frac{a^8}{a} $ as $ \frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a} $ - one 'a' cancels out, leaving 7 a's!

Q: What if the bottom exponent is bigger than the top?

You still subtract! For example, $ \frac{a^3}{a^5} = a^{3-5} = a^{-2} $. The negative exponent means one over that positive power: $ \frac{1}{a^2} $.

Q: Do I need to remember 'a' equals 'a¹'?

Yes! Any variable without a visible exponent has an implied exponent of 1. So $ a = a^1 $, which is why we subtract 1 from 8.

Q: Can I use this rule with numbers too?

Absolutely! $ \frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25 $. The rule works for any base as long as you're dividing powers of the same number or variable.

Q: What happens when the exponents are equal?

When exponents are equal, like $ \frac{a^3}{a^3} $, you get $ a^{3-3} = a^0 = 1 $. Any non-zero number to the power of 0 equals 1!

Exponent Rules with Division Operations

Insert the corresponding expression:

$\frac{a^8}{a}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 Any number to the power of 1 equals itself

00:08 Let's use the formula for dividing powers

00:10 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:13 equals the number (A) to the power of the difference of exponents (M-N)

00:16 Let's use this formula in our exercise

00:18 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:

$\frac{a^8}{a}=$

Step-by-step solution

To solve the expression $\frac{a^8}{a}$ , we can use the Power of a Quotient Rule for Exponents. According to this rule, when dividing like bases, we subtract the exponents.

The general formula for this is:

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

For the given expression:

$m = 8$
$n = 1$

Now, applying the formula:

$\frac{a^8}{a} = a^{8-1} = a^7$

Therefore, the solution to the question is:

$a^7$

Final Answer

$a^7$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^8}{a} = a^{8-1} = a^7$
Check: Verify by expanding: $\frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a} = a^7$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add 8 + 1 = 9 to get $a^9$ ! Addition rule only applies to multiplication, not division. Always subtract the bottom exponent from the top exponent when dividing powers with the same base.

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?

When you divide powers, you're essentially canceling out common factors. Think of $\frac{a^8}{a}$ as $\frac{a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a}{a}$ - one 'a' cancels out, leaving 7 a's!

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{a^3}{a^5} = a^{3-5} = a^{-2}$ . The negative exponent means one over that positive power: $\frac{1}{a^2}$ .

Do I need to remember 'a' equals 'a¹'?

Yes! Any variable without a visible exponent has an implied exponent of 1. So $a = a^1$ , which is why we subtract 1 from 8.

Can I use this rule with numbers too?

Absolutely! $\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25$ . The rule works for any base as long as you're dividing powers of the same number or variable.

What happens when the exponents are equal?

When exponents are equal, like $\frac{a^3}{a^3}$ , you get $a^{3-3} = a^0 = 1$ . Any non-zero number to the power of 0 equals 1!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime