Evaluate (11×5×4)/(9×13×17) Raised to the Eighth Power

Q: Why does each number need to be raised to the 8th power?

Because of the power rule for products: $ (a \times b)^n = a^n \times b^n $. When you have multiple factors like $ 11 \times 5 \times 4 $, the exponent must be applied to each individual factor.

Q: What's the difference between the numerator and denominator here?

The numerator is $ 11 \times 5 \times 4 $ and becomes $ 11^8 \times 5^8 \times 4^8 $. The denominator is $ 9 \times 13 \times 17 $ and becomes $ 9^8 \times 13^8 \times 17^8 $. Both follow the same power rule!

Q: How do I remember which rule to use?

Think of it as distributing the exponent! Just like you distribute multiplication over addition, you distribute the exponent over every factor in the product. The key is: don't leave anyone out!

Q: What if I accidentally apply the power to just the whole fraction?

You'd get $ \frac{(11 \times 5 \times 4)^8}{9 \times 13 \times 17} $, which is incorrect! The denominator also needs the exponent. Always remember: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $.

Power of Products with Fraction Bases

Insert the corresponding expression:

$\left(\frac{11\times5\times4}{9\times13\times17}\right)^8=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction raised to the power (N)

00:09 equals the numerator and denominator with the same power (N)

00:13 Note that both numerator and denominator are products

00:16 We will apply this formula to our exercise

00:27 According to the laws of exponents when a product is raised to the power (N)

00:31 it is equal to each factor in the product separately raised to the same power (N)

00:39 We will apply this formula to our exercise

01:00 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:

$\left(\frac{11\times5\times4}{9\times13\times17}\right)^8=$

Step-by-step solution

To solve the problem, follow these steps:

Step 1: Use the property $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ to distribute the exponent 8 to both the numerator and the denominator.
Step 2: Raise each component in the numerator and the denominator to the power of 8.

Start by applying the rule to the entire fraction:

$\left(\frac{11\times5\times4}{9\times13\times17}\right)^8 = \frac{(11\times5\times4)^8}{(9\times13\times17)^8}$

Then apply the rule of powers to the product inside both the numerator and denominator:

$= \frac{11^8 \times 5^8 \times 4^8}{9^8 \times 13^8 \times 17^8}$

Therefore, the solution to the given problem is:

$\frac{11^8\times5^8\times4^8}{9^8\times13^8\times17^8}$

Final Answer

$\frac{11^8\times5^8\times4^8}{9^8\times13^8\times17^8}$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising a fraction to a power, raise both numerator and denominator to that power
Technique: Apply $(a \times b)^n = a^n \times b^n$ to each factor individually
Check: Verify each factor has the same exponent: $11^8, 5^8, 4^8$ and $9^8, 13^8, 17^8$ ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only some factors
Don't raise just one number to the 8th power like $\frac{11 \times 5 \times 4^8}{9 \times 13 \times 17}$ = completely wrong result! This violates the power rules and creates an unequal expression. Always apply the exponent to every single factor in both numerator and denominator.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does each number need to be raised to the 8th power?

Because of the power rule for products: $(a \times b)^n = a^n \times b^n$ . When you have multiple factors like $11 \times 5 \times 4$ , the exponent must be applied to each individual factor.

Can I just multiply 11×5×4 first, then raise it to the 8th power?

Yes! Both methods give the same answer. You could calculate $220^8$ and $1989^8$ , but keeping the factors separate as $11^8 \times 5^8 \times 4^8$ is usually easier to work with.

What's the difference between the numerator and denominator here?

The numerator is $11 \times 5 \times 4$ and becomes $11^8 \times 5^8 \times 4^8$ . The denominator is $9 \times 13 \times 17$ and becomes $9^8 \times 13^8 \times 17^8$ . Both follow the same power rule!

How do I remember which rule to use?

Think of it as distributing the exponent! Just like you distribute multiplication over addition, you distribute the exponent over every factor in the product. The key is: don't leave anyone out!

What if I accidentally apply the power to just the whole fraction?

You'd get $\frac{(11 \times 5 \times 4)^8}{9 \times 13 \times 17}$ , which is incorrect! The denominator also needs the exponent. Always remember: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

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