Solve for (3/4)^x: Complete the Expression Problem

Q: Why can't I just multiply the numbers by x like in option 2?

Because exponentiation is different from multiplication! $ \left(\frac{3}{4}\right)^x $ means multiply $ \frac{3}{4} $ by itself x times, not multiply 3 and 4 by x.

Q: How do I remember to apply the exponent to both parts?

Think of it as distribution: the exponent outside the parentheses gets distributed to every part inside. Just like distributing multiplication, but with exponents!

Q: What if the exponent was a specific number like 3?

$ \left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64} $. The same rule applies whether the exponent is a variable or a specific number.

Q: Can I simplify this expression further?

Not without knowing the value of x! $ \frac{3^x}{4^x} $ is the simplest form when x is a variable. You could also write it as $ \left(\frac{3}{4}\right)^x $.

Q: What's wrong with the other answer choices?

$ \frac{3x}{4x} $ treats the exponent like multiplication$ \frac{3^x}{4} $ only applies the exponent to the numerator$ \frac{3}{4^x} $ only applies the exponent to the denominator

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{3}{4}\right)^x=$

❤️ 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

00:06 Let's simplify the following problem.

00:09 Remember, with exponents, a fraction to the power of N

00:14 means both the top and bottom are each raised to N.

00:18 Let's apply this rule to our exercise.

00:21 And here is how we solve it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(\frac{3}{4}\right)^x=$

Step-by-step solution

To solve this problem, we will apply the rules of exponents:

Step 1: Identify the Given Expression
Step 2: Apply the Exponent Rule

Now, let's work through these steps:

Step 1: We are given the expression $\left(\frac{3}{4}\right)^x$ .

Step 2: According to the rule of exponents, when a fraction is raised to a power, this is equivalent to raising both the numerator and the denominator to that power. Therefore, we have:

$\left(\frac{3}{4}\right)^x = \frac{3^x}{4^x}$

Therefore, the expression $\left(\frac{3}{4}\right)^x$ is equivalent to $\frac{3^x}{4^x}$ .

Thus, the correct answer is option 1, which is $\frac{3^x}{4^x}$ .

The solution to the problem is $\frac{3^x}{4^x}$ .

Final Answer

$\frac{3^x}{4^x}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: When raising a fraction to a power, apply exponent to both numerator and denominator
Technique: $\left(\frac{3}{4}\right)^x = \frac{3^x}{4^x}$ by distributing the exponent
Check: Verify by expanding: $\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2} = \frac{9}{16}$ ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the numerator
Don't write $\left(\frac{3}{4}\right)^x = \frac{3^x}{4}$ = wrong result! This ignores the denominator and violates exponent rules. Always apply the exponent to both the numerator AND denominator when raising a fraction to a power.

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 multiply the numbers by x like in option 2?

Because exponentiation is different from multiplication! $\left(\frac{3}{4}\right)^x$ means multiply $\frac{3}{4}$ by itself x times, not multiply 3 and 4 by x.

How do I remember to apply the exponent to both parts?

Think of it as distribution: the exponent outside the parentheses gets distributed to every part inside. Just like distributing multiplication, but with exponents!

What if the exponent was a specific number like 3?

$\left(\frac{3}{4}\right)^3 = \frac{3^3}{4^3} = \frac{27}{64}$ . The same rule applies whether the exponent is a variable or a specific number.

Can I simplify this expression further?

Not without knowing the value of x! $\frac{3^x}{4^x}$ is the simplest form when x is a variable. You could also write it as $\left(\frac{3}{4}\right)^x$ .

What's wrong with the other answer choices?

$\frac{3x}{4x}$ treats the exponent like multiplication
$\frac{3^x}{4}$ only applies the exponent to the numerator
$\frac{3}{4^x}$ only applies the exponent to the denominator

🌟 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