Solve (5×11)/(3×7) Raised to Power a: Complex Fraction Expression

Q: Why can't I just raise one number in each product to the power a?

Because the exponent applies to the entire product, not individual factors. When you have $ (5\times11)^a $, the exponent a affects both 5 and 11 equally.

Q: Are both answer choices A and C really correct?

Yes! Choice A shows $ \frac{(5\times11)^a}{(3\times7)^a} $ and Choice C shows $ \frac{5^a\times11^a}{3^a\times7^a} $. These are equivalent expressions using different exponent rules.

Q: What's the difference between the power of a fraction rule and product power rule?

The power of a fraction rule $ \left(\frac{x}{y}\right)^a = \frac{x^a}{y^a} $ applies to the entire fraction. The product power rule $ (xy)^a = x^a \times y^a $ applies within the numerator and denominator separately.

Q: Can I simplify 5×11 and 3×7 before applying the exponent?

You could calculate $ 5\times11 = 55 $ and $ 3\times7 = 21 $ to get $ \left(\frac{55}{21}\right)^a $, but the question asks for the expression form, not the simplified numerical result.

Exponent Rules with Complex Fractions

Insert the corresponding expression:

$\left(\frac{5\times11}{3\times7}\right)^a=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following exercise

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)

00:12 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)

00:27 it is equal to each factor in the product separately raised to the same power (N)

00:31 We will apply this formula to our exercise

00:37 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:

$\left(\frac{5\times11}{3\times7}\right)^a=$

Step-by-step solution

To solve this problem, follow these steps:

Identify the given expression: $\left(\frac{5 \times 11}{3 \times 7}\right)^a$ .
Apply the exponent rule for powers of a fraction: $\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$ .
Apply this rule separately to the numerator and the denominator:

The numerator is $5 \times 11$ and the denominator is $3 \times 7$ . When the fraction is raised to a power $a$ , we apply the power to both the numerator and denominator:

$\left(\frac{5 \times 11}{3 \times 7}\right)^a = \frac{(5 \times 11)^a}{(3 \times 7)^a}$

Which corresponds to option 1.

Each product is raised to the power $a$ . By exponent rules $(xy)^a = x^a \times y^a$ , this expression becomes:

$\frac{5^a \times 11^a}{3^a \times 7^a}$

Thus, the expression can be rewritten as: $\frac{5^a \times 11^a}{3^a \times 7^a}$ .

Referring to the provided choices, this matches choice 3.

Therefore, the correct choice is 4, A+C are correct.

Final Answer

A'+C' are correct

Key Points to Remember

Essential concepts to master this topic

Power of a Fraction Rule: $\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$ applies to entire fraction
Product Power Rule: $(xy)^a = x^a \times y^a$ applies to each factor separately
Verification: Check both forms are equivalent: $\frac{(5\times11)^a}{(3\times7)^a} = \frac{5^a\times11^a}{3^a\times7^a}$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only some factors
Don't raise just one number to power a like $\frac{5\times11^a}{3\times7^a}$ = wrong distribution! This ignores the exponent rules and creates incorrect expressions. Always apply the exponent to every single factor when using $(xy)^a = x^a \times y^a$ .

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 raise one number in each product to the power a?

Because the exponent applies to the entire product, not individual factors. When you have $(5\times11)^a$ , the exponent a affects both 5 and 11 equally.

Are both answer choices A and C really correct?

Yes! Choice A shows $\frac{(5\times11)^a}{(3\times7)^a}$ and Choice C shows $\frac{5^a\times11^a}{3^a\times7^a}$ . These are equivalent expressions using different exponent rules.

How do I know when to expand the exponents further?

It depends on what the problem asks for! Sometimes you keep products together like $(5\times11)^a$ , other times you expand using $(xy)^a = x^a \times y^a$ . Both forms are mathematically correct.

What's the difference between the power of a fraction rule and product power rule?

The power of a fraction rule $\left(\frac{x}{y}\right)^a = \frac{x^a}{y^a}$ applies to the entire fraction. The product power rule $(xy)^a = x^a \times y^a$ applies within the numerator and denominator separately.

Can I simplify 5×11 and 3×7 before applying the exponent?

You could calculate $5\times11 = 55$ and $3\times7 = 21$ to get $\left(\frac{55}{21}\right)^a$ , but the question asks for the expression form, not the simplified numerical result.

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