https://file.aiquickdraw.com/mathbot/file/17270158800536trt5pk9.jpg

The expression given in the image is:

\[
\frac{3x^5}{7x^4} \div \frac{6x^{-6}}{14x^{-4}}
\]

Let's solve this step by step.

### Step 1: Rewrite the division as multiplication by the reciprocal

Division of fractions is equivalent to multiplying by the reciprocal of the second fraction:

\[
\frac{3x^5}{7x^4} \times \frac{14x^{-4}}{6x^{-6}}
\]

### Step 2: Simplify the coefficients

Now, simplify the constants (numerical coefficients):
\[
\frac{3}{7} \times \frac{14}{6}
\]
\[
\frac{3}{7} \times \frac{14}{6} = \frac{3 \times 14}{7 \times 6} = \frac{42}{42} = 1
\]

So the constants simplify to 1.

### Step 3: Simplify the powers of \(x\)

Using the rule of exponents \( \frac{x^a}{x^b} = x^{a-b} \):

For the first fraction:
\[
\frac{x^5}{x^4} = x^{5-4} = x^1
\]

For the second fraction:
\[
\frac{x^{-4}}{x^{-6}} = x^{-4 - (-6)} = x^{-4 + 6} = x^2
\]

### Step 4: Multiply the results

Now, multiply the simplified expressions:

\[
x^1 \times x^2 = x^{1+2} = x^3
\]

### Final Answer:
The simplified expression is:
\[
x^3
\]

Would you like any further details or clarification?

Here are 5 related questions:
1. What are the rules of exponents used to simplify algebraic expressions?
2. How do you divide fractions, and what does it mean to multiply by the reciprocal?
3. Can you explain how to handle negative exponents when simplifying expressions?
4. How would the problem change if one of the terms had a negative coefficient?
5. How do you simplify complex fractions with variables?

**Tip:** Always start by simplifying numerical coefficients and then handle the variables using the exponent rules.

Simplify the following algebraic expression: (3x^5 / 7x^4) ÷ (6x^-6 / 14x^-4).

Learn how to simplify the algebraic expression (3x^5 / 7x^4) ÷ (6x^-6 / 14x^-4) using exponent rules and fraction division techniques. This step-by-step solution is ideal for students in Grades 8-10, focusing on algebraic operations involving exponents.

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