Simplify the Expression: 8^16 Divided by 8^8

Q: Why do we subtract exponents when dividing?

Think of it as canceling out matching factors! $ \frac{8^{16}}{8^8} $ means you have 16 eights multiplied together on top, and 8 eights on bottom. The 8 eights cancel out, leaving you with 16 - 8 = 8 eights still multiplied together.

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

You still subtract! For example, $ \frac{8^3}{8^5} = 8^{3-5} = 8^{-2} $. The negative exponent means you'll have a fraction as your final answer.

Q: Can I use this rule with different bases like 8 and 2?

No! This rule only works when the bases are exactly the same. For $ \frac{8^{16}}{2^8} $, you'd need to convert one base to match the other first.

Q: How do I remember when to add vs subtract exponents?

Easy memory trick: Multiplication = Add, Division = Subtract. When you multiply $ 8^a \times 8^b $, you add exponents. When you divide $ \frac{8^a}{8^b} $, you subtract!

Q: What does 8^8 actually equal in regular numbers?

While $ 8^8 $ equals 16,777,216, most problems want you to leave the answer in exponential form since it's simpler and shows the mathematical relationship clearly.

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{8^{16}}{8^8}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's begin.

00:09 We'll use a special formula for dividing powers.

00:13 When a number, like A, to the power of N is divided by A to the power of M.

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

00:23 Let's try using this formula in our exercise.

00:27 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{8^{16}}{8^8}=$

Step-by-step solution

The given expression is $\frac{8^{16}}{8^8}$ . To solve this, we apply the Power of a Quotient Rule for Exponents.

This rule states that when dividing two exponential expressions with the same base, we subtract the exponent of the denominator from the exponent of the numerator. Mathematically, it can be expressed as:

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

In this problem, the base $8$ is the same in both the numerator and the denominator, so we can apply this rule.

Subtract the exponent of the denominator from the exponent of the numerator:

$16 - 8 = 8$

Therefore, the simplified form of the given expression is:

$8^8$

Thus, the answer is $8^8$ .

Final Answer

$8^8$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents
Technique: $\frac{8^{16}}{8^8} = 8^{16-8} = 8^8$
Check: Verify that $8^8 \times 8^8 = 8^{16}$ using multiplication rule ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add exponents like 16 + 8 = 24 for division = $8^{24}$ ! Addition rule only applies to multiplication. Always subtract the bottom exponent from the top exponent when dividing same bases.

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 matching factors! $\frac{8^{16}}{8^8}$ means you have 16 eights multiplied together on top, and 8 eights on bottom. The 8 eights cancel out, leaving you with 16 - 8 = 8 eights still multiplied together.

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{8^3}{8^5} = 8^{3-5} = 8^{-2}$ . The negative exponent means you'll have a fraction as your final answer.

Can I use this rule with different bases like 8 and 2?

No! This rule only works when the bases are exactly the same. For $\frac{8^{16}}{2^8}$ , you'd need to convert one base to match the other first.

How do I remember when to add vs subtract exponents?

Easy memory trick: Multiplication = Add, Division = Subtract. When you multiply $8^a \times 8^b$ , you add exponents. When you divide $\frac{8^a}{8^b}$ , you subtract!

What does 8^8 actually equal in regular numbers?

While $8^8$ equals 16,777,216, most problems want you to leave the answer in exponential form since it's simpler and shows the mathematical relationship clearly.

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