Solve (3×4)/(5×11×9)^y: Finding the Equivalent Expression

Q: Why are both answer choices A and B correct?

Both expressions are mathematically equivalent! Choice A shows $ \frac{3^y\times4^y}{5^y\times11^y\times9^y} $ (fully distributed), while choice B shows $ \frac{(3\times4)^y}{(5\times11\times9)^y} $ (partially distributed). They represent the same value.

Q: When do I use the power of a fraction rule?

Use $ \left(\frac{a}{b}\right)^c = \frac{a^c}{b^c} $ whenever you see a fraction raised to a power. This rule lets you move the exponent inside to both numerator and denominator.

Q: What's the difference between the two correct forms?

The difference is how far you distribute the exponent. Form A distributes completely to individual factors, while form B keeps some factors grouped. Both are valid stopping points!

Q: Do I always need to distribute the exponent completely?

Not always! Sometimes keeping factors grouped (like choice B) is cleaner. The key is applying power rules correctly at each step, whether you stop early or go all the way.

Q: How do I know which form the question wants?

Read carefully! This question asks for the corresponding expression, meaning any mathematically equivalent form. When multiple forms are correct, look for an option like 'A+B are correct'.

Power Rules with Fraction Expressions

Insert the corresponding expression:

$\left(\frac{3\times4}{5\times11\times9}\right)^y=$

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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)

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

00:11 Note that both the numerator and denominator are products

00:14 We will apply this formula to our exercise

00:22 According to the laws of exponents when a 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)

00:29 We will apply this formula to our exercise

00:44 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:

$\left(\frac{3\times4}{5\times11\times9}\right)^y=$

Step-by-step solution

To simplify the given expression, we start with the original expression:

$\left(\frac{3\times4}{5\times11\times9}\right)^y$ .

Using the property for powers of a fraction, we distribute the exponent $y$ to the numerator and the denominator:

$\left(\frac{a}{b}\right)^c = \frac{a^c}{b^c}$

First, apply the formula:

$\left(\frac{3\times4}{5\times11\times9}\right)^y = \frac{(3\times4)^y}{(5\times11\times9)^y}$ .

Next, apply the power of a product property, $(ab)^c = a^c \times b^c$ , to both the numerator and the denominator:

The numerator becomes $(3\times4)^y = 3^y \times 4^y$ .

The denominator becomes $(5\times11\times9)^y = 5^y \times 11^y \times 9^y$ .

Thus, the fully simplified expression is:

$\frac{3^y\times4^y}{5^y\times11^y\times9^y}$ .

After comparing with the given options, this matches choice 1 and 2, so option 4 is the right one: A+B are correct

Final Answer

a'+b' are correct

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $(ab)^y = a^y \times b^y$ distributes over multiplication
Check: Verify both forms are equivalent by expanding step-by-step ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't apply y to just one number like $3 \times 4^y$ = wrong form! This violates the power of a product rule. Always distribute the exponent to every factor: $(3 \times 4)^y = 3^y \times 4^y$ .

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why are both answer choices A and B correct?

Both expressions are mathematically equivalent! Choice A shows $\frac{3^y\times4^y}{5^y\times11^y\times9^y}$ (fully distributed), while choice B shows $\frac{(3\times4)^y}{(5\times11\times9)^y}$ (partially distributed). They represent the same value.

When do I use the power of a fraction rule?

Use $\left(\frac{a}{b}\right)^c = \frac{a^c}{b^c}$ whenever you see a fraction raised to a power. This rule lets you move the exponent inside to both numerator and denominator.

What's the difference between the two correct forms?

The difference is how far you distribute the exponent. Form A distributes completely to individual factors, while form B keeps some factors grouped. Both are valid stopping points!

Do I always need to distribute the exponent completely?

Not always! Sometimes keeping factors grouped (like choice B) is cleaner. The key is applying power rules correctly at each step, whether you stop early or go all the way.

How do I know which form the question wants?

Read carefully! This question asks for the corresponding expression, meaning any mathematically equivalent form. When multiple forms are correct, look for an option like 'A+B are correct'.

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