Solve (5/9)^(2x+1): Complete the Exponential Expression

Q: What if the exponent is just a number like 3?

The same rule applies! $ \left(\frac{5}{9}\right)^3 = \frac{5^3}{9^3} = \frac{125}{729} $. Whether the exponent is a number or an expression like 2x+1, both parts get the same exponent.

Q: How do I know which answer choice is correct?

Look for the choice where both the numerator and denominator have exactly the same exponent. In this case, both should have $ 2x+1 $ as the exponent.

Q: Can I simplify the exponent 2x+1 first?

No need to! The exponent $ 2x+1 $ should stay exactly as it is. The power rule works with any exponent, whether it's simple or complex.

Exponential Expressions with Fractional Bases

Insert the corresponding expression:

$\left(\frac{5}{9}\right)^{2x+1}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:04 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 Note that each exponent (N) contains an addition operation

00:14 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:

$\left(\frac{5}{9}\right)^{2x+1}=$

Step-by-step solution

To solve this problem, we'll apply the power of a fraction rule, which states that $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Step 1: Recognize the given expression: $\left(\frac{5}{9}\right)^{2x+1}$ .

Step 2: Apply the exponent rule to rewrite the expression. According to the rule, this becomes:

$\left(\frac{5}{9}\right)^{2x+1} = \frac{5^{2x+1}}{9^{2x+1}}$ .

Therefore, the expression can be rewritten as $\frac{5^{2x+1}}{9^{2x+1}}$ .

Among the given choices, the correct option is choice 1: $\frac{5^{2x+1}}{9^{2x+1}}$ .

Thus, the solution to the problem is $\frac{5^{2x+1}}{9^{2x+1}}$ .

Final Answer

$\frac{5^{2x+1}}{9^{2x+1}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: Transform $\left(\frac{5}{9}\right)^{2x+1}$ to $\frac{5^{2x+1}}{9^{2x+1}}$
Check: Verify both parts have identical exponents: 2x+1 appears twice ✓

Common Mistakes

Avoid these frequent errors

Adding exponents to numerator and denominator
Don't write $\frac{5^{2x}}{9^{2x}}+\frac{5}{9}$ by treating addition like distribution = completely wrong structure! This confuses addition with exponentiation rules. Always apply the same exponent to both numerator and denominator: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

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 the exponent?

Exponents don't work like multiplication! When you raise a fraction to a power, you must raise both parts separately. Think of $\left(\frac{5}{9}\right)^3$ as $\frac{5}{9} \times \frac{5}{9} \times \frac{5}{9} = \frac{5^3}{9^3}$ .

What if the exponent is just a number like 3?

The same rule applies! $\left(\frac{5}{9}\right)^3 = \frac{5^3}{9^3} = \frac{125}{729}$ . Whether the exponent is a number or an expression like 2x+1, both parts get the same exponent.

How do I know which answer choice is correct?

Look for the choice where both the numerator and denominator have exactly the same exponent. In this case, both should have $2x+1$ as the exponent.

Can I simplify the exponent 2x+1 first?

No need to! The exponent $2x+1$ should stay exactly as it is. The power rule works with any exponent, whether it's simple or complex.

What's wrong with the other answer choices?

Choice 2 incorrectly adds terms instead of applying exponents
Choice 3 only applies the exponent to the numerator
Choice 4 only applies the exponent to the denominator

Only choice 1 correctly applies the exponent to both parts!

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