Simplify the Power Fraction: (10^5)/(17^5) Expression

Q: Why can't I just cancel out the exponent 5 from top and bottom?

Exponents don't cancel like regular numbers! The rule $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $ keeps the exponent but combines the bases. Think of it as factoring out the common power.

Q: How do I remember which exponent rule to use?

When you see identical exponents in numerator and denominator, use $ \frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n $. When bases are the same, use $ \frac{a^m}{a^n} = a^{m-n} $.

Q: What if the numbers were different, like 10³/17⁵?

Then you cannot use this rule! The exponents must be exactly the same to combine them into $ \left(\frac{a}{b}\right)^n $. Different exponents require different approaches.

Q: Is there a way to check if my answer is correct?

Yes! Calculate both expressions as decimals: $ \frac{10^5}{17^5} \approx 0.0007 $ and $ \left(\frac{10}{17}\right)^5 \approx 0.0007 $. They should be exactly equal.

Q: Why does this rule work mathematically?

Think of it as factoring: $ \frac{10^5}{17^5} = \frac{10 \times 10 \times 10 \times 10 \times 10}{17 \times 17 \times 17 \times 17 \times 17} $. You can group each pair of 10/17 together, giving you five copies of 10/17 multiplied together!

Exponent Rules with Power Division

Insert the corresponding expression:

$\frac{10^5}{17^5}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, a fraction raised to the power (N)

00:08 equals the numerator and denominator, each raised to the same power (N)

00:12 We'll apply this formula to our exercise, only this time in the opposite direction

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{10^5}{17^5}=$

Step-by-step solution

To solve the given problem, we want to rewrite the expression $\frac{10^5}{17^5}$ using the rules of exponents.

Step 1: Recognize that both the numerator and the denominator are raised to the 5th power.
Step 2: Apply the rule $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , which allows us to combine the power into a single expression.

By applying this rule, we have:

$\frac{10^5}{17^5} = \left(\frac{10}{17}\right)^5$

This shows that the original expression can be rewritten as a single power of a fraction.

Therefore, the simplified form of the expression is $\left(\frac{10}{17}\right)^5$ .

Final Answer

$\left(\frac{10}{17}\right)^5$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, use $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$
Technique: Combine identical exponents: $\frac{10^5}{17^5} = \left(\frac{10}{17}\right)^5$
Check: Expand both forms to verify they give same decimal value ✓

Common Mistakes

Avoid these frequent errors

Trying to divide the exponents instead of the bases
Don't change $\frac{10^5}{17^5}$ into something like $\frac{10}{17}$ by "canceling" the 5s! This completely ignores the exponent rule and gives a wildly different answer. Always apply the rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ to keep the same exponent.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just cancel out the exponent 5 from top and bottom?

Exponents don't cancel like regular numbers! The rule $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ keeps the exponent but combines the bases. Think of it as factoring out the common power.

How do I remember which exponent rule to use?

When you see identical exponents in numerator and denominator, use $\frac{a^n}{b^n} = \left(\frac{a}{b}\right)^n$ . When bases are the same, use $\frac{a^m}{a^n} = a^{m-n}$ .

What if the numbers were different, like 10³/17⁵?

Then you cannot use this rule! The exponents must be exactly the same to combine them into $\left(\frac{a}{b}\right)^n$ . Different exponents require different approaches.

Is there a way to check if my answer is correct?

Yes! Calculate both expressions as decimals: $\frac{10^5}{17^5} \approx 0.0007$ and $\left(\frac{10}{17}\right)^5 \approx 0.0007$ . They should be exactly equal.

Why does this rule work mathematically?

Think of it as factoring: $\frac{10^5}{17^5} = \frac{10 \times 10 \times 10 \times 10 \times 10}{17 \times 17 \times 17 \times 17 \times 17}$ . You can group each pair of 10/17 together, giving you five copies of 10/17 multiplied together!

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