Complete the Expression: (11/19) Raised to (a+3b) Power

Q: Can I simplify a+3b in the exponent first?

No! Since we don't know the values of a and b, we must keep the exponent as $ a+3b $. Don't try to split it up - treat the whole expression as one exponent.

Q: How is this different from (11/19)^a + (11/19)^(3b)?

Those are completely different! $ \left(\frac{11}{19}\right)^{a+3b} $ means one fraction raised to the power of a+3b, while your expression would be adding two separate powers together.

Q: What's wrong with putting the exponent only on 11?

Writing $ \frac{11^{a+3b}}{19} $ means you only raised the numerator to the power, not the whole fraction! Remember: the exponent applies to the entire base, which is the fraction $ \frac{11}{19} $.

Exponential Expressions with Fraction Bases

Insert the corresponding expression:

$\left(\frac{11}{19}\right)^{a+3b}=$

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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

00:15 Note that the exponent (N) contains an addition operation

00:19 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:

$\left(\frac{11}{19}\right)^{a+3b}=$

Step-by-step solution

To solve this problem, we need to rewrite the expression $\left(\frac{11}{19}\right)^{a+3b}$ using the rules for powers of fractions. Specifically, we apply the exponent to both the numerator and the denominator separately.

According to the rule $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$ , we apply the exponent $a+3b$ to both 11 and 19:

$\left(\frac{11}{19}\right)^{a+3b} = \frac{11^{a+3b}}{19^{a+3b}}$

Therefore, the expression can be rewritten as $\frac{11^{a+3b}}{19^{a+3b}}$ , matching choice 3 in the provided options.

Hence, the solution to the problem is $\frac{11^{a+3b}}{19^{a+3b}}$ .

Final Answer

$\frac{11^{a+3b}}{19^{a+3b}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{11}{19}\right)^{a+3b} = \frac{11^{a+3b}}{19^{a+3b}}$
Check: Both top and bottom have identical exponent $a+3b$ ✓

Common Mistakes

Avoid these frequent errors

Distributing exponent incorrectly to fraction
Don't write $\frac{11^a}{19^a}+3b$ or split the exponent = wrong structure! This treats addition in the exponent as separate terms instead of one power. Always apply the entire exponent $a+3b$ to both numerator and denominator as a single unit.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why does the exponent apply to both the top and bottom?

When you raise a fraction to a power, you're multiplying that fraction by itself multiple times. So $\left(\frac{11}{19}\right)^2 = \frac{11}{19} \times \frac{11}{19} = \frac{11^2}{19^2}$ . The same pattern works for any exponent!

What if the exponent is a+3b instead of just a number?

It doesn't matter if the exponent is a simple number or an expression like $a+3b$ ! The rule $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$ works for any exponent, whether it's 2, 5, or a+3b.

Can I simplify a+3b in the exponent first?

No! Since we don't know the values of a and b, we must keep the exponent as $a+3b$ . Don't try to split it up - treat the whole expression as one exponent.

How is this different from (11/19)^a + (11/19)^(3b)?

Those are completely different! $\left(\frac{11}{19}\right)^{a+3b}$ means one fraction raised to the power of a+3b, while your expression would be adding two separate powers together.

What's wrong with putting the exponent only on 11?

Writing $\frac{11^{a+3b}}{19}$ means you only raised the numerator to the power, not the whole fraction! Remember: the exponent applies to the entire base, which is the fraction $\frac{11}{19}$ .

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