Simplify the Expression: (7²x²)/a² Algebraic Fraction

Q: Why can I combine the squares like this?

When you have all terms raised to the same power, you can factor them together! The rule $ \frac{b^m \times c^m}{d^m} = \left(\frac{bc}{d}\right)^m $ works because multiplication and division distribute over exponents.

Q: What if not all terms have the same exponent?

Then you cannot use this factoring rule! You can only combine terms when they have exactly the same exponent. Mixed exponents require different approaches.

Q: How do I know which form is 'simpler'?

Both forms are mathematically equal, but $ \left(\frac{7x}{a}\right)^2 $ is often preferred because it shows the structure more clearly - it's obviously a perfect square of a rational expression.

Q: Can I expand this back to check my work?

Absolutely! $ \left(\frac{7x}{a}\right)^2 = \frac{(7x)^2}{a^2} = \frac{7^2 \times x^2}{a^2} $. If you get back to the original expression, you know your factoring was correct!

Q: What about the coefficient 7 - does it stay inside the parentheses?

Yes! The 7 is squared just like the x, so it must stay inside. Writing $ 7 \times \left(\frac{x}{a}\right)^2 $ would mean only the x and a are squared, not the 7.

Exponent Properties with Algebraic Fractions

Insert the corresponding expression:

$\frac{7^2\times x^2}{a^2}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 product that is raised to the power (N)

00:06 is equal to the product broken down into factors where each factor is raised to the power (N)

00:11 We will apply this formula to our exercise, in the reverse direction

00:14 We will combine the factors into a multiplication operation within parentheses

00:20 According to the laws of exponents, a fraction that is raised to the power (N)

00:24 equals a fraction where both the numerator and the denominator are raised to the power (N)

00:28 We will apply this formula to our exercise, in the reverse direction

00:32 We will place the entire fraction inside of parentheses and raise it to the appropriate power

00:36 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:

$\frac{7^2\times x^2}{a^2}=$

Step-by-step solution

Let's solve the problem step by step:

We start with the expression:

$\frac{7^2 \times x^2}{a^2}$ .

Recognizing the terms in the expression, we notice that:

$7^2$ is the square of $7$ .
$x^2$ is the square of $x$ .
$a^2$ is the square of $a$ .

Using one of the properties of exponents, we know that:

$\frac{b^m \times c^m}{d^m} = \left(\frac{b \times c}{d}\right)^m$ .

Thus, we can rewrite our given expression:

$\frac{7^2 \times x^2}{a^2}$ can be rewritten as $\left(\frac{7 \times x}{a}\right)^2$ .

This conversion works because the squares in the numerator and the denominator allow us to apply the rule of powers over fractions.

The equivalent expression is therefore $\left(\frac{7 \times x}{a}\right)^2$ .

Final Answer

$\left(\frac{7\times x}{a}\right)^2$

Key Points to Remember

Essential concepts to master this topic

Rule: Perfect squares in numerator and denominator can be factored together
Technique: Transform $\frac{7^2 \times x^2}{a^2}$ to $\left(\frac{7x}{a}\right)^2$
Check: Expand result back: $\left(\frac{7x}{a}\right)^2 = \frac{49x^2}{a^2}$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent rules incorrectly to mixed terms
Don't write $7 \times \left(\frac{x}{a}\right)^2$ = different structure! This separates the 7 from the squared terms, creating an entirely different expression. Always group all squared terms together: $\frac{7^2 \times x^2}{a^2} = \left(\frac{7x}{a}\right)^2$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine the squares like this?

When you have all terms raised to the same power, you can factor them together! The rule $\frac{b^m \times c^m}{d^m} = \left(\frac{bc}{d}\right)^m$ works because multiplication and division distribute over exponents.

What if not all terms have the same exponent?

Then you cannot use this factoring rule! You can only combine terms when they have exactly the same exponent. Mixed exponents require different approaches.

How do I know which form is 'simpler'?

Both forms are mathematically equal, but $\left(\frac{7x}{a}\right)^2$ is often preferred because it shows the structure more clearly - it's obviously a perfect square of a rational expression.

Can I expand this back to check my work?

Absolutely! $\left(\frac{7x}{a}\right)^2 = \frac{(7x)^2}{a^2} = \frac{7^2 \times x^2}{a^2}$ . If you get back to the original expression, you know your factoring was correct!

What about the coefficient 7 - does it stay inside the parentheses?

Yes! The 7 is squared just like the x, so it must stay inside. Writing $7 \times \left(\frac{x}{a}\right)^2$ would mean only the x and a are squared, not the 7.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime