Complete the Expression: (2/3) Raised to Power a

Q: Why can't I just multiply the fraction by itself a times?

You absolutely can! When you multiply $ \frac{2}{3} \times \frac{2}{3} \times ... $ (a times), you get $ \frac{2 \times 2 \times ...}{3 \times 3 \times ...} = \frac{2^a}{3^a} $. The exponent rule is just a shortcut for this repeated multiplication!

Q: What's wrong with writing 2a/3a as the answer?

Exponentiation is not the same as multiplication! Writing $ \frac{2a}{3a} $ means "2 times a over 3 times a", which equals 2/3. But $ \left(\frac{2}{3}\right)^a $ means repeated multiplication, not simple multiplication by a.

Q: Does this rule work for negative exponents too?

Yes! For example, $ \left(\frac{2}{3}\right)^{-2} = \frac{2^{-2}}{3^{-2}} = \frac{1/4}{1/9} = \frac{9}{4} $. The same rule applies regardless of whether the exponent is positive, negative, or even a fraction!

Q: What if the fraction in the problem was improper, like 5/3?

The rule works exactly the same way! $ \left(\frac{5}{3}\right)^a = \frac{5^a}{3^a} $. It doesn't matter if the fraction is proper, improper, or mixed - the exponent rule always applies to both parts.

Fractional Exponentiation with Variable Powers

Insert the corresponding expression:

$\left(\frac{2}{3}\right)^a=$

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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)

00:11 We will apply this formula to our exercise

00:16 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:

$\left(\frac{2}{3}\right)^a=$

Step-by-step solution

Let's determine the corresponding expression for $\left(\frac{2}{3}\right)^a$ :

We apply the property of exponentiation for fractions, which states:
$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$ .

Substituting $x = 2$ , $y = 3$ , and $n = a$ , we have:

$\left(\frac{2}{3}\right)^a = \frac{2^a}{3^a}$ .

Therefore, the correct expression is $\frac{2^a}{3^a}$ .

Assessing the possible choices:

Choice 1: $\frac{2}{3^a}$ - This is incorrect as it does not raise the numerator to $a$ .
Choice 2: $\frac{2a}{3a}$ - This is incorrect as it misuses the exponent rule.
Choice 3: $\frac{2^a}{3^a}$ - This is correct, as it follows the exponentiation property.
Choice 4: $\frac{2^a}{3}$ - This is incorrect as it does not raise the denominator to $a$ .

Thus, the correct choice is Choice 3: $\frac{2^a}{3^a}$ .

Final Answer

$\frac{2^a}{3^a}$

Key Points to Remember

Essential concepts to master this topic

Rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ applies the exponent to both numerator and denominator
Technique: For $\left(\frac{2}{3}\right)^a$ , raise 2 to power a and 3 to power a separately
Check: Both numerator and denominator must have the same exponent a ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only numerator or denominator
Don't write $\frac{2^a}{3}$ or $\frac{2}{3^a}$ = incomplete application! This violates the fundamental exponent rule for fractions. Always apply the exponent to both the numerator AND denominator.

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 fraction by itself a times?

You absolutely can! When you multiply $\frac{2}{3} \times \frac{2}{3} \times ...$ (a times), you get $\frac{2 \times 2 \times ...}{3 \times 3 \times ...} = \frac{2^a}{3^a}$ . The exponent rule is just a shortcut for this repeated multiplication!

What's wrong with writing 2a/3a as the answer?

Exponentiation is not the same as multiplication! Writing $\frac{2a}{3a}$ means "2 times a over 3 times a", which equals 2/3. But $\left(\frac{2}{3}\right)^a$ means repeated multiplication, not simple multiplication by a.

Does this rule work for negative exponents too?

Yes! For example, $\left(\frac{2}{3}\right)^{-2} = \frac{2^{-2}}{3^{-2}} = \frac{1/4}{1/9} = \frac{9}{4}$ . The same rule applies regardless of whether the exponent is positive, negative, or even a fraction!

Can I simplify 2^a/3^a further?

Usually no, unless you know specific values for a. However, you could write it as $\left(\frac{2}{3}\right)^a$ , which is actually the original form! Both expressions are equivalent and correct.

What if the fraction in the problem was improper, like 5/3?

The rule works exactly the same way! $\left(\frac{5}{3}\right)^a = \frac{5^a}{3^a}$ . It doesn't matter if the fraction is proper, improper, or mixed - the exponent rule always applies to both parts.

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