Evaluate (5×6×7)/(9×11×13) Raised to Power b: Complex Fraction Expression

Q: Why does every number get the exponent b?

When you raise a product to a power, the power distributes to each factor. Think of it like this: $ (2 \times 3)^2 = (2 \times 3) \times (2 \times 3) = 2^2 \times 3^2 $.

Q: What if I forget to apply the exponent to all factors?

You'll get the wrong answer! The power rule for products says every factor gets the exponent. Missing even one factor breaks the mathematical rule.

Q: Does this work the same way for the denominator?

Yes! The power rule applies to both numerator and denominator. So $ (9 \times 11 \times 13)^b = 9^b \times 11^b \times 13^b $.

Q: Can I simplify the fraction before applying the exponent?

You could, but it's usually easier to apply the exponent first. The power rule works directly with products, so use it as shown in this problem.

Q: What happens if b = 1?

When b = 1, you get the original fraction back! Every number to the power of 1 equals itself: $ 5^1 = 5 $, $ 6^1 = 6 $, etc.

Q: Is there a shortcut for this type of problem?

The power rule for fractions is your shortcut: $ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} $. Just remember to apply it to every factor in the products!

Power Rules with Complex Fractions

Insert the corresponding expression:

$\left(\frac{5\times6\times7}{9\times11\times13}\right)^b=$

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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 raised to the same power (N)

00:13 Note that both numerator and denominator are products

00:17 We will apply this formula to our exercise

00:22 According to the laws of exponents when the entire product is raised to the power (N)

00:26 it is equal to each factor in the product separately raised to the same power (N)

00:32 We will apply this formula to our exercise

00:48 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:

$\left(\frac{5\times6\times7}{9\times11\times13}\right)^b=$

Step-by-step solution

To solve this problem, we'll use exponentiation properties to simplify the given expression:

Step 1: Apply the exponent to each term in the numerator: $(5 \times 6 \times 7)^b = 5^b \times 6^b \times 7^b$ .
Step 2: Apply the exponent to each term in the denominator: $(9 \times 11 \times 13)^b = 9^b \times 11^b \times 13^b$ .
Step 3: Use the property of exponents for fractions: $\left(\frac{5 \times 6 \times 7}{9 \times 11 \times 13}\right)^b = \frac{5^b \times 6^b \times 7^b}{9^b \times 11^b \times 13^b}$ .

As we have followed the rules of exponents and simplified accordingly, the corresponding expression is:

$\frac{5^b \times 6^b \times 7^b}{9^b \times 11^b \times 13^b}$ .

Final Answer

$\frac{5^b\times6^b\times7^b}{9^b\times11^b\times13^b}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Raise each factor in numerator and denominator to b
Technique: $(5 \times 6 \times 7)^b = 5^b \times 6^b \times 7^b$
Check: Each original number appears with exponent b in final answer ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to some factors
Don't raise just 7 to power b while leaving 5×6 unchanged = $\frac{5 \times 6 \times 7^b}{9 \times 11 \times 13}$ ! This violates the power rule for products. 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 every number get the exponent b?

When you raise a product to a power, the power distributes to each factor. Think of it like this: $(2 \times 3)^2 = (2 \times 3) \times (2 \times 3) = 2^2 \times 3^2$ .

What if I forget to apply the exponent to all factors?

You'll get the wrong answer! The power rule for products says every factor gets the exponent. Missing even one factor breaks the mathematical rule.

Does this work the same way for the denominator?

Yes! The power rule applies to both numerator and denominator. So $(9 \times 11 \times 13)^b = 9^b \times 11^b \times 13^b$ .

Can I simplify the fraction before applying the exponent?

You could, but it's usually easier to apply the exponent first. The power rule works directly with products, so use it as shown in this problem.

What happens if b = 1?

When b = 1, you get the original fraction back! Every number to the power of 1 equals itself: $5^1 = 5$ , $6^1 = 6$ , etc.

Is there a shortcut for this type of problem?

The power rule for fractions is your shortcut: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ . Just remember to apply it to every factor in the products!

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