Simplify 60^60 ÷ 60^42: Power Division Problem

Q: Why do I subtract exponents when dividing?

Think of it this way: $ 60^{60} ÷ 60^{42} $ means you have 60 multiplied by itself 60 times, then you're dividing by 60 multiplied by itself 42 times. The 42 factors of 60 in the bottom cancel out 42 factors from the top, leaving you with $ 60^{60-42} = 60^{18} $.

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

You still subtract! For example, $ 5^3 ÷ 5^7 = 5^{3-7} = 5^{-4} $. The negative exponent means you have a fraction: $ \frac{1}{5^4} $.

Q: Can I use this rule with different bases?

No! This rule only works when the bases are exactly the same. You cannot simplify $ \frac{60^{60}}{30^{42}} $ using this method because 60 ≠ 30.

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

Use this memory trick: Multiplication = ADD exponentsDivision = SUBTRACT exponentsThink: "More multiplication, more addition!"

Exponent Division with Same Base

Insert the corresponding expression:

$\frac{60^{60}}{60^{42}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's break this down step by step.

00:11 We will use the formula for dividing powers.

00:15 When you have a number A raised to the power of N, divided by the same base A raised to the power of M,

00:21 it equals A to the power of M minus N. So simple!

00:27 Let's apply this formula to solve our exercise.

00:31 And that's how you find the solution to the question. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{60^{60}}{60^{42}}=$

Step-by-step solution

To solve the expression $\frac{60^{60}}{60^{42}}$ , we need to apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases, we subtract the exponents. In mathematical terms, for any non-zero number $a$ , and integers $m$ and $n$ , $\frac{a^m}{a^n} = a^{m-n}$ .

Applying this rule to our problem:

We have the same base: $60$ .
We subtract the exponent in the denominator from the exponent in the numerator: $60^{60-42}$ .
This simplifies the expression to $60^{18}$ .

Therefore, the solution to the question is: $60^{18}$ .

Final Answer

$60^{18}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents: $a^m ÷ a^n = a^{m-n}$
Technique: Calculate $60^{60-42} = 60^{18}$ by subtracting exponents
Check: Verify base stays same and exponents subtract correctly: 60 - 42 = 18 ✓

Common Mistakes

Avoid these frequent errors

Adding or multiplying exponents instead of subtracting
Don't add exponents like $60^{60+42} = 60^{102}$ or multiply them like $60^{60×42}$ ! Division requires subtraction, not addition or multiplication. Always remember: division of same bases means subtract the bottom exponent from the top exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract exponents when dividing?

Think of it this way: $60^{60} ÷ 60^{42}$ means you have 60 multiplied by itself 60 times, then you're dividing by 60 multiplied by itself 42 times. The 42 factors of 60 in the bottom cancel out 42 factors from the top, leaving you with $60^{60-42} = 60^{18}$ .

What if the bottom exponent is bigger than the top?

You still subtract! For example, $5^3 ÷ 5^7 = 5^{3-7} = 5^{-4}$ . The negative exponent means you have a fraction: $\frac{1}{5^4}$ .

Can I use this rule with different bases?

No! This rule only works when the bases are exactly the same. You cannot simplify $\frac{60^{60}}{30^{42}}$ using this method because 60 ≠ 30.

How do I remember when to add vs subtract exponents?

Use this memory trick:

Multiplication = ADD exponents
Division = SUBTRACT exponents

Think: "More multiplication, more addition!"

Is there a way to check my answer without a calculator?

Yes! You can verify the exponent arithmetic: 60 - 42 = 18, so your answer should be $60^{18}$ . Also check that your base didn't change - it should still be 60.

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