Simplify the Power Expression: 5^7 ÷ 5^10

Q: Why do I subtract 10 from 7 and not 7 from 10?

With division $ \frac{a^m}{a^n} $, you always subtract the denominator exponent from the numerator exponent: m - n. So it's 7 - 10, which gives us the negative exponent $ 5^{-3} $.

Q: What does a negative exponent mean?

A negative exponent means "one over" the positive version. So $ 5^{-3} = \frac{1}{5^3} = \frac{1}{125} $. It's a way to write very small fractions!

Q: Can I just calculate 5^7 ÷ 5^10 directly?

You could, but that means calculating $ 78,125 ÷ 9,765,625 $ - huge numbers! Using the exponent rule $ 5^{7-10} = 5^{-3} $ is much faster and cleaner.

Q: How do I know which answer choice is correct?

Look for the choice that shows subtraction in the right order: numerator exponent minus denominator exponent. That's $ 5^{7-10} $, not $ 5^{10-7} $!

Q: Is 5^(-3) the same as -5^3?

No! $ 5^{-3} = \frac{1}{125} $ (positive fraction), but $ -5^3 = -125 $ (negative whole number). The negative sign's position makes a huge difference!

Exponent Division with Negative Results

Insert the corresponding expression:

$\frac{5^7}{5^{10}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 Let's use the formula for dividing powers

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:06 equals the number (A) to the power of the difference of exponents (M-N)

00:08 Let's use this formula in our exercise

00:10 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:

$\frac{5^7}{5^{10}}=$

Step-by-step solution

To solve the expression $\frac{5^7}{5^{10}}$ , we need to apply the Power of a Quotient Rule for Exponents. This rule states that when dividing like bases, you subtract the exponents:

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

In this particular case, the base is 5, and the exponents are 7 and 10. Using the rule, we subtract the exponent in the denominator from the exponent in the numerator:

Numerator exponent = 7
Denominator exponent = 10

Therefore, we get:

5^{7-10}

.

In conclusion, the simplified form of the given expression is:

5^{-3}

The solution to the question is: $5^{7-10}$ .

Final Answer

$5^{7-10}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing same bases, subtract the exponents
Technique: $\frac{5^7}{5^{10}} = 5^{7-10} = 5^{-3}$
Check: Convert to fraction: $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when dividing
Don't add 7 + 10 = 17 when dividing powers = $5^{17}$ ! This gives a huge positive number instead of a tiny fraction. Always subtract the bottom exponent from the top exponent in division.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I subtract 10 from 7 and not 7 from 10?

With division $\frac{a^m}{a^n}$ , you always subtract the denominator exponent from the numerator exponent: m - n. So it's 7 - 10, which gives us the negative exponent $5^{-3}$ .

What does a negative exponent mean?

A negative exponent means "one over" the positive version. So $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$ . It's a way to write very small fractions!

Can I just calculate 5^7 ÷ 5^10 directly?

You could, but that means calculating $78,125 ÷ 9,765,625$ - huge numbers! Using the exponent rule $5^{7-10} = 5^{-3}$ is much faster and cleaner.

How do I know which answer choice is correct?

Look for the choice that shows subtraction in the right order: numerator exponent minus denominator exponent. That's $5^{7-10}$ , not $5^{10-7}$ !

Is 5^(-3) the same as -5^3?

No! $5^{-3} = \frac{1}{125}$ (positive fraction), but $-5^3 = -125$ (negative whole number). The negative sign's position makes a huge difference!

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