Complete the Expression: (3×7)/(4×8) Raised to Power (b+1)

Q: Why does the exponent (b+1) apply to both 3 and 7?

Because of the power of a product rule: $ (a \times b)^n = a^n \times b^n $. When you raise a multiplication to a power, each factor gets raised to that power.

Q: Can I simplify 3×7 and 4×8 first?

You could get $ \left(\frac{21}{32}\right)^{b+1} $, but the question asks for the expanded form showing each base raised to the power separately.

Q: How do I remember the exponent rules?

Think: "Exponents distribute over multiplication". Just like distributing in algebra, when you have $ (a \times b)^n $, the exponent n visits each factor!

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

Same rule applies! $ \left(\frac{3 \times 7}{4 \times 8}\right)^3 = \frac{3^3 \times 7^3}{4^3 \times 8^3} $. The exponent always distributes to every factor in the fraction.

Exponent Rules with Fractional Bases

Insert the corresponding expression:

$\left(\frac{3\times7}{4\times8}\right)^{b+1}=$

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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 a power (N)

00:08 equals the numerator and denominator raised to the same power (N)

00:13 We will apply this formula to our exercise

00:21 According to the laws of exponents when a product is raised to a power (N)

00:26 it is equal to each factor in the product separately raised to the same power (N)

00:33 We will apply this formula to our exercise

00:47 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:

$\left(\frac{3\times7}{4\times8}\right)^{b+1}=$

Step-by-step solution

To solve the expression $\left(\frac{3\times7}{4\times8}\right)^{b+1}$ , we follow these steps:

Step 1: Apply the power $(b+1)$ to the entire fraction, $\left(\frac{3 \times 7}{4 \times 8}\right)^{b+1}$ .
Step 2: Use the exponentiation rule $(\frac{a}{b})^n = \frac{a^n}{b^n}$ to apply the power to both numerator and denominator separately.
Step 3: Expand the numerator and the denominator: $(3 \times 7)^{b+1} = 3^{b+1} \times 7^{b+1}$ and $(4 \times 8)^{b+1} = 4^{b+1} \times 8^{b+1}$ .
Step 4: Combine these results into one fraction: $\frac{3^{b+1} \times 7^{b+1}}{4^{b+1} \times 8^{b+1}}$ .

Therefore, the simplified expression is $\frac{3^{b+1}\times7^{b+1}}{4^{b+1}\times8^{b+1}}$ .

Upon examining the choices, the correct option is choice 3: $\frac{3^{b+1}\times7^{b+1}}{4^{b+1}\times8^{b+1}}$ .

Final Answer

$\frac{3^{b+1}\times7^{b+1}}{4^{b+1}\times8^{b+1}}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $(a \times b)^n = a^n \times b^n$ for each term
Check: Verify each base gets the same exponent $(b+1)$ ✓

Common Mistakes

Avoid these frequent errors

Applying exponent to only one number in each product
Don't raise just 3 to power (b+1) and leave 7 unchanged = $\frac{3^{b+1} \times 7}{4^{b+1} \times 8}$ ! This ignores the multiplication rule for exponents. Always apply the exponent to every factor when raising a product to a power.

Practice Quiz

Test your knowledge with interactive questions

$$ Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

FAQ

Everything you need to know about this question

Why does the exponent (b+1) apply to both 3 and 7?

Because of the power of a product rule: $(a \times b)^n = a^n \times b^n$ . When you raise a multiplication to a power, each factor gets raised to that power.

What's the difference between the answer choices?

Choice 1 incorrectly uses different exponents (b vs 1). Choice 2 only applies the exponent to one number in each product. Only choice 3 correctly applies (b+1) to all four numbers.

Can I simplify 3×7 and 4×8 first?

You could get $\left(\frac{21}{32}\right)^{b+1}$ , but the question asks for the expanded form showing each base raised to the power separately.

How do I remember the exponent rules?

Think: "Exponents distribute over multiplication". Just like distributing in algebra, when you have $(a \times b)^n$ , the exponent n visits each factor!

What if the exponent was just a number like 3?

Same rule applies! $\left(\frac{3 \times 7}{4 \times 8}\right)^3 = \frac{3^3 \times 7^3}{4^3 \times 8^3}$ . The exponent always distributes to every factor in the fraction.

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