Simplify (4/ab)²: Complete the Squared Fraction Expression

Q: Why do I need parentheses around the denominator?

The parentheses show that the exponent applies to the entire denominator $ ab $. Without them, you might only square one variable, which gives a completely different answer!

Q: Do I have to simplify 4² to 16?

Not necessarily! Both $ \frac{4^2}{(ab)^2} $ and $ \frac{16}{a^2b^2} $ are correct. The question asks for the form that matches the given choices.

Q: What's the difference between (ab)² and ab²?

Huge difference! $ (ab)^2 = a^2b^2 $ squares both variables, while $ ab^2 $ only squares b. Always use parentheses when squaring multiple terms.

Q: How do I remember the exponent rule for fractions?

Think: "Power on top, power on bottom". The exponent outside affects both the numerator and denominator: $ \left(\frac{m}{n}\right)^p = \frac{m^p}{n^p} $

Q: Can I distribute the exponent differently?

No! The exponent must be applied to the entire fraction first. You can't split it up or apply it to individual parts without following the proper exponent rules.

Exponent Rules with Fractional Expressions

Insert the corresponding expression:

$\left(\frac{4}{a\times b}\right)^2=$

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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 a 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

00:17 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:

$\left(\frac{4}{a\times b}\right)^2=$

Step-by-step solution

To solve the problem, let's apply exponent rules to the given expression:

Step 1: Identify the fraction's components. The numerator is $4$ and the denominator is $a \times b$ .
Step 2: Apply the exponent rule for fractions, $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$ . In this case, $m = 4$ , $n = a \times b$ , and $p = 2$ .

Now, we apply the exponent:

$\left(\frac{4}{a \times b}\right)^2 = \frac{4^2}{(a \times b)^2}$ .

This results in:

$\frac{16}{a^2 \times b^2}$ .

However, the expression $\frac{4^2}{(a \times b)^2}$ matches choice 2 from the provided options, hence:

The correct answer to the problem in its intended form is $\frac{4^2}{\left(a\times b\right)^2}$ .

Final Answer

$\frac{4^2}{\left(a\times b\right)^2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponents to both numerator and denominator separately
Technique: $\left(\frac{4}{ab}\right)^2 = \frac{4^2}{(ab)^2} = \frac{16}{a^2b^2}$
Check: Verify parentheses apply to entire denominator: $(ab)^2 = a^2b^2$ ✓

Common Mistakes

Avoid these frequent errors

Only applying exponent to part of denominator
Don't apply the exponent to just one variable like $\frac{4^2}{ab^2}$ = wrong answer! The exponent outside parentheses must apply to the entire expression inside. Always apply the exponent to the complete denominator: $(ab)^2 = a^2b^2$ .

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do I need parentheses around the denominator?

The parentheses show that the exponent applies to the entire denominator $ab$ . Without them, you might only square one variable, which gives a completely different answer!

Do I have to simplify 4² to 16?

Not necessarily! Both $\frac{4^2}{(ab)^2}$ and $\frac{16}{a^2b^2}$ are correct. The question asks for the form that matches the given choices.

What's the difference between (ab)² and ab²?

Huge difference! $(ab)^2 = a^2b^2$ squares both variables, while $ab^2$ only squares b. Always use parentheses when squaring multiple terms.

How do I remember the exponent rule for fractions?

Think: "Power on top, power on bottom". The exponent outside affects both the numerator and denominator: $\left(\frac{m}{n}\right)^p = \frac{m^p}{n^p}$

Can I distribute the exponent differently?

No! The exponent must be applied to the entire fraction first. You can't split it up or apply it to individual parts without following the proper exponent rules.

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