Simplify the Expression: 17^16 ÷ 17^20 Using Laws of Exponents

Exponent Division with Negative Results

Insert the corresponding expression:

17161720= \frac{17^{16}}{17^{20}}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:10 Let's start with something simple.
00:12 We'll use the formula for dividing powers.
00:16 Take any number A raised to the power of N.
00:20 Divide it by the same base, A, raised to the power of M.
00:25 This equals A raised to the power of M minus N.
00:29 We're using this for our exercise.
00:32 And that's how we find the solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

17161720= \frac{17^{16}}{17^{20}}=

2

Step-by-step solution

To solve the expression 17161720 \frac{17^{16}}{17^{20}} , we can apply the Power of a Quotient Rule for Exponents. This rule states that when you divide two exponents with the same base, you can subtract the exponents to simplify the expression.

The given expression is:

17161720 \frac{17^{16}}{17^{20}}

According to the Quotient Rule for Exponents, this expression can be simplified as:

1716−20 17^{16-20}

Here's the step-by-step breakdown:

  • The base of both the numerator and the denominator is the same, that is, 17.
  • According to the rule, subtract the exponent in the denominator from the exponent in the numerator: 16−20 16 - 20 .
  • This gives us the exponent: −4 -4 .

So, the simplified expression is:

17−4 17^{-4}

However, as requested, we should express this as:

1716−20 17^{16-20}

The solution to the question is:

1716−20 17^{16-20}

3

Final Answer

1716−20 17^{16-20}

Key Points to Remember

Essential concepts to master this topic
  • Quotient Rule: When dividing same bases, subtract the exponents
  • Technique: 17161720=1716−20=17−4 \frac{17^{16}}{17^{20}} = 17^{16-20} = 17^{-4}
  • Check: Verify by expanding: 17161720=1174 \frac{17^{16}}{17^{20}} = \frac{1}{17^4} ✓

Common Mistakes

Avoid these frequent errors
  • Adding instead of subtracting exponents
    Don't add exponents when dividing = 1716+20=1736 17^{16+20} = 17^{36} which is completely wrong! This confuses multiplication rules with division. 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?

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Think of it like canceling! 17161720 \frac{17^{16}}{17^{20}} means you have 16 factors of 17 on top and 20 on bottom. After canceling 16 pairs, you're left with 4 factors of 17 in the denominator, which gives 17−4 17^{-4} .

What does a negative exponent mean?

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A negative exponent means "one over" the positive version! So 17−4=1174 17^{-4} = \frac{1}{17^4} . It's not a negative number - it's a fraction!

Should I calculate the actual number value?

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Usually no - leave it as 17−4 17^{-4} or 1716−20 17^{16-20} ! The question asks for the expression, not the decimal value. Computing 174 17^4 would give a very large number anyway.

How do I remember the quotient rule?

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Use the phrase "Same base, subtract the race!" When bases match, subtract the exponents. Think: aman=am−n \frac{a^m}{a^n} = a^{m-n} (top minus bottom).

What if the bottom exponent is bigger than the top?

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That's exactly what happens here! When the denominator's exponent (20) is larger than the numerator's (16), you get a negative exponent: 16−20=−4 16 - 20 = -4 . This is perfectly normal!

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