Simplify the Complex Fraction: (8⁷ × 13⁶)/(19⁶ × 3⁷)

Q: Why can't I just multiply all the numbers together first?

Because the exponents are different (7 and 6)! You can only combine terms when they have the same exponent. Think of $ 8^7 $ and $ 13^6 $ as completely different types of numbers.

Q: What's the difference between the quotient rule and product rule?

The quotient rule is for division: $ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m $. The product rule is for multiplication: $ a^m \times a^n = a^{m+n} $. Here we use quotient rule because we're dividing powers.

Q: Can I simplify this fraction to a decimal?

While you could calculate the decimal, it's better to leave it in exact form as $ \left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6 $. This shows your understanding of exponent rules and is the preferred mathematical answer.

Q: How do I know which terms to group together?

Look at the exponents! Group terms that have the same exponent: $ 8^7 $ with $ 3^7 $, and $ 13^6 $ with $ 19^6 $. Same exponents can use the quotient rule together.

Q: What if the exponents were all the same?

Great question! If all terms had the same exponent, like $ \frac{8^5 \times 13^5}{19^5 \times 3^5} $, then you could write it as $ \left(\frac{8 \times 13}{19 \times 3}\right)^5 $. But here, 7 ≠ 6, so we keep them separate.

Exponent Properties with Complex Fractions

Insert the corresponding expression:

$\frac{8^7\times13^6}{19^6\times3^7}=$

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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 exponent laws, a fraction raised to the power (N)

00:07 equals a fraction where both the numerator and denominator are raised to the power (N)

00:11 We'll apply this formula to our exercise, in the reverse direction

00:15 We'll break it down into 2 fractions with appropriate powers

00:19 We'll place each fraction inside of parentheses and raise it to the appropriate power

00:25 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:

$\frac{8^7\times13^6}{19^6\times3^7}=$

Step-by-step solution

To solve the problem, we apply the properties of exponents and simplify each component of the fraction separately.

Step 1: Analyze the expression.

The given expression is $\frac{8^7\times13^6}{19^6\times3^7}$ .
We need to simplify this using exponent rules.

Step 2: Apply the rule of exponents to simplify each pair of terms.

Simplify the expression $\frac{8^7}{3^7}$ using the rule $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ to get $\left(\frac{8}{3}\right)^7$ .
Simplify $\frac{13^6}{19^6}$ to get $\left(\frac{13}{19}\right)^6$ .

Step 3: Combine the simplified components into a single expression.

The expression becomes $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$ .

choice 1 : The simplified expression matches choice $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$ , which means this is the correct answer.

Therefore, the simplified expression is $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$ .

Final Answer

$\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When bases have same exponent, use $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$
Technique: Group terms with matching exponents: $\frac{8^7}{3^7} \times \frac{13^6}{19^6}$
Check: Verify exponents match in final form: 7 and 6 appear once each ✓

Common Mistakes

Avoid these frequent errors

Combining all terms under one exponent
Don't write $\left(\frac{8 \times 13}{19 \times 3}\right)^{13}$ by adding exponents = wrong powers! The exponents 7 and 6 are different, so terms can't be combined this way. Always group terms with matching exponents separately using the quotient rule.

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 multiply all the numbers together first?

Because the exponents are different (7 and 6)! You can only combine terms when they have the same exponent. Think of $8^7$ and $13^6$ as completely different types of numbers.

What's the difference between the quotient rule and product rule?

The quotient rule is for division: $\frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$ . The product rule is for multiplication: $a^m \times a^n = a^{m+n}$ . Here we use quotient rule because we're dividing powers.

Can I simplify this fraction to a decimal?

While you could calculate the decimal, it's better to leave it in exact form as $\left(\frac{8}{3}\right)^7\times\left(\frac{13}{19}\right)^6$ . This shows your understanding of exponent rules and is the preferred mathematical answer.

How do I know which terms to group together?

Look at the exponents! Group terms that have the same exponent: $8^7$ with $3^7$ , and $13^6$ with $19^6$ . Same exponents can use the quotient rule together.

What if the exponents were all the same?

Great question! If all terms had the same exponent, like $\frac{8^5 \times 13^5}{19^5 \times 3^5}$ , then you could write it as $\left(\frac{8 \times 13}{19 \times 3}\right)^5$ . But here, 7 ≠ 6, so we keep them separate.

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