Complete the Expression: (ax/7)^6 Algebraic Fraction Power

Q: Why does the exponent 6 apply to both a and x?

Because of the power of a product rule! When you have $ (a \times x)^6 $, the exponent applies to the entire product, so each factor gets raised to the 6th power separately.

Q: Can I simplify this expression further?

Not really! $ \frac{a^6 \times x^6}{7^6} $ is already in its simplest form. You could write it as $ \frac{a^6x^6}{7^6} $ (without the multiplication symbol), but that's just notation.

Q: How do I remember which rule to use first?

Start with the fraction power rule to separate numerator and denominator, then apply the product power rule to handle multiple factors. Work from outside in!

Fractional Exponents with Product Powers

Insert the corresponding expression:

$\left(\frac{a\times x}{7}\right)^6=$

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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 laws of exponents, a fraction raised to the power (N)

00:07 equals the numerator and denominator raised to the same power (N)

00:12 We will apply this formula to our exercise

00:17 According to the laws of exponents when a product is raised to the power (N)

00:21 it is equal to each factor in the product separately raised to the same power (N)

00:26 We will apply this formula to our exercise

00:32 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:

$\left(\frac{a\times x}{7}\right)^6=$

Step-by-step solution

To solve this problem, we'll utilize the properties of exponents to simplify the expression $\left(\frac{a \times x}{7}\right)^6$ .

Let's proceed with the steps:

Step 1: Utilize the exponent rule for fractions: $\left( \frac{m}{n} \right)^p = \frac{m^p}{n^p}$ . This allows us to express the given expression as:

$\left(\frac{a \times x}{7}\right)^6 = \frac{(a \times x)^6}{7^6}$

Step 2: Apply the exponent rule to the multinomial in the numerator: $(a \times x)^6 = a^6 \times x^6$ .

Thus, we have:

$\frac{(a \times x)^6}{7^6} = \frac{a^6 \times x^6}{7^6}$

Conclusion: The correct expression is $\frac{a^6 \times x^6}{7^6}$ , which corresponds to choice 4.

Final Answer

$\frac{a^6\times x^6}{7^6}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Product Rule: $(a \times x)^6 = a^6 \times x^6$ distributes to each factor
Check: Verify each variable has the same exponent as original power ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one factor in the numerator
Don't write $(a \times x)^6 = a^6 \times x$ or $a \times x^6$ = wrong answer! This ignores the power rule for products. Always distribute the exponent to every single factor: $(a \times x)^6 = a^6 \times x^6$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the exponent 6 apply to both a and x?

Because of the power of a product rule! When you have $(a \times x)^6$ , the exponent applies to the entire product, so each factor gets raised to the 6th power separately.

What happens to the 7 in the denominator?

The 7 also gets raised to the 6th power! Using the fraction power rule $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$ , we get $7^6$ in the denominator.

Can I simplify this expression further?

Not really! $\frac{a^6 \times x^6}{7^6}$ is already in its simplest form. You could write it as $\frac{a^6x^6}{7^6}$ (without the multiplication symbol), but that's just notation.

What if I had different exponents in the original problem?

The same rules apply! Whatever exponent is outside the parentheses gets applied to every factor inside. For example: $\left(\frac{ab}{3}\right)^4 = \frac{a^4b^4}{3^4}$

How do I remember which rule to use first?

Start with the fraction power rule to separate numerator and denominator, then apply the product power rule to handle multiple factors. Work from outside in!

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