Simplify the Expression: 5^9 ÷ 5^9 in Fraction Form

Exponent Division with Same Base

Insert the corresponding expression:

5959 \frac{5^9}{5^9}

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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 the formula for dividing powers. Are you ready?
00:14 When a number, A, to the power of N, is divided by A to the power of M,
00:20 it equals A to the power of M minus N. Simple, right?
00:25 We'll apply this formula in our exercise now.
00:28 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

5959 \frac{5^9}{5^9}

2

Step-by-step solution

To solve this problem, we apply the Power of a Quotient Rule for exponents. This rule is applicable when both the numerator and the denominator of a fraction have the same base. The rule states:


aman=amn \frac{a^m}{a^n} = a^{m-n}


For our problem, the expression is:


5959 \frac{5^9}{5^9}


In this expression, the base a a is 5 5 , m m is 9 9 , and n n is also 9 9 . We apply the rule as follows:


5959=599 \frac{5^9}{5^9} = 5^{9-9}


Calculating the exponent:


99=0 9 - 9 = 0


So the expression becomes:


50 5^0


Any number raised to the power of 0 is 1, but in this context, we are simply reducing the original expression to its simplest form. Therefore, 50 5^0 is the correct answer.


The solution to the question is: 50 5^0

3

Final Answer

50 5^0

Key Points to Remember

Essential concepts to master this topic
  • Rule: When dividing powers with same base, subtract the exponents
  • Technique: 5959=599=50 \frac{5^9}{5^9} = 5^{9-9} = 5^0
  • Check: Any number to the zero power equals 1, so 50=1 5^0 = 1

Common Mistakes

Avoid these frequent errors
  • Dividing the exponents instead of subtracting
    Don't divide 9 ÷ 9 = 1 to get 51 5^1 ! This ignores the quotient rule and gives the wrong answer. Always subtract exponents: 599=50 5^{9-9} = 5^0 .

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why is 50 5^0 equal to 1?

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The zero exponent rule states that any non-zero number raised to the power of 0 equals 1. Think of it as: 5959=1 \frac{5^9}{5^9} = 1 because any number divided by itself is 1!

Do I always subtract exponents when dividing?

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Only when the bases are the same! For aman \frac{a^m}{a^n} , subtract to get amn a^{m-n} . If bases are different, you can't use this rule.

What if the bottom exponent is bigger?

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You still subtract! If you have 5257=527=55 \frac{5^2}{5^7} = 5^{2-7} = 5^{-5} . Negative exponents mean you flip the fraction: 155 \frac{1}{5^5} .

Can I just cancel out the 59 5^9 terms?

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Yes, but be careful! Canceling is another way to think about it, but writing 599=50 5^{9-9} = 5^0 shows you understand the exponent rule and is mathematically more precise.

Is this the same as 5959=11=1 \frac{5^9}{5^9} = \frac{1}{1} = 1 ?

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Absolutely! Both methods give the same result. The exponent rule gives us 50 5^0 , which equals 1. It's just a more algebraic way to express the answer.

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