Calculate (6×8)⁴: Evaluating a Product Raised to Fourth Power

Q: Why are both $ 6^4 \times 8^4 $ and $ 48^4 $ correct?

Both expressions represent the same calculation! Option (a) uses the power of product rule to distribute the exponent: $ (6 \times 8)^4 = 6^4 \times 8^4 $. Option (b) first calculates the product inside parentheses: $ 6 \times 8 = 48 $, then raises it to the fourth power: $ 48^4 $.

Q: When should I use the power of product rule versus calculating first?

It depends on the situation! Use the power of product rule when you need to show the expanded form or when the numbers get too large. Calculate first when you want the simplest numerical form or when doing mental math.

Q: Does the power of product rule work with more than two factors?

Yes! For example, $ (2 \times 3 \times 5)^3 = 2^3 \times 3^3 \times 5^3 $. The exponent distributes to every factor inside the parentheses, no matter how many there are.

Q: What's the difference between this and the power of power rule?

The power of product rule deals with $ (a \times b)^n $, while the power of power rule deals with $ (a^m)^n = a^{mn} $. Don't confuse multiplying bases with multiplying exponents!

Q: Why can't I write $ (6 \times 8)^4 = 6^4 \times 8 $ ?

This violates the power of product rule! When you have $ (a \times b)^n $, the exponent must apply to both factors. Writing $ 6^4 \times 8 $ only raises 6 to the fourth power, leaving 8 unchanged.

Exponent Rules with Product Operations

Choose the expression that corresponds to the following:

$\left(6\times8\right)^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this problem together.

00:10 To open parentheses with an exponent, remember this tip.

00:16 Raise each number inside to the power.

00:19 We'll use this handy formula in our example.

00:23 This is one solution. Now, let's try solving it another way.

00:31 First, multiply inside the parentheses, then raise to the power.

00:38 And there you have it, the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$\left(6\times8\right)^4=$

Step-by-step solution

To solve the problem of rewriting the expression $(6 \times 8)^4$ , we will apply the power of a product rule. This rule states that $(a \times b)^n = a^n \times b^n$ .

Let's apply this rule to our expression:

$(6 \times 8)^4$ can be rewritten as $6^4 \times 8^4$ by applying the Power of a Product rule.

Now, let's consider the available options:

Choice 1: $6^4 \times 8^4$ - This choice directly corresponds to our rewritten expression using the power of a product rule.
Choice 2: $48^4$ - This represents the product calculated first ( $6 \times 8 = 48$ ) and then raised to the fourth power. This is also a valid representation since $(6 \times 8)^4$ is equivalent to $48^4$ .
Choice 3: $6^4 \times 8$ - This is incorrect because it does not correctly apply the power of a product rule.

Therefore, the correct answer according to the given choices is that (a) and (b) are correct.

Final Answer

Answers (a) and (b) are correct.

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: $(a \times b)^n = a^n \times b^n$ distributes the exponent
Technique: Apply rule to get $(6 \times 8)^4 = 6^4 \times 8^4$
Check: Both $6^4 \times 8^4$ and $48^4$ equal 5,308,416 ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to one factor
Don't write $(6 \times 8)^4 = 6^4 \times 8$ = wrong result! This only raises 6 to the fourth power while leaving 8 unchanged, which violates the power of product rule. Always apply the exponent to every factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why are both $6^4 \times 8^4$ and $48^4$ correct?

Both expressions represent the same calculation! Option (a) uses the power of product rule to distribute the exponent: $(6 \times 8)^4 = 6^4 \times 8^4$ . Option (b) first calculates the product inside parentheses: $6 \times 8 = 48$ , then raises it to the fourth power: $48^4$ .

When should I use the power of product rule versus calculating first?

It depends on the situation! Use the power of product rule when you need to show the expanded form or when the numbers get too large. Calculate first when you want the simplest numerical form or when doing mental math.

Does the power of product rule work with more than two factors?

Yes! For example, $(2 \times 3 \times 5)^3 = 2^3 \times 3^3 \times 5^3$ . The exponent distributes to every factor inside the parentheses, no matter how many there are.

What's the difference between this and the power of power rule?

The power of product rule deals with $(a \times b)^n$ , while the power of power rule deals with $(a^m)^n = a^{mn}$ . Don't confuse multiplying bases with multiplying exponents!

Why can't I write $(6 \times 8)^4 = 6^4 \times 8$ ?

This violates the power of product rule! When you have $(a \times b)^n$ , the exponent must apply to both factors. Writing $6^4 \times 8$ only raises 6 to the fourth power, leaving 8 unchanged.

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Calculate (6×8)⁴: Evaluating a Product Raised to Fourth Power

Exponent Rules with Product Operations

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why are both $6^4 \times 8^4$ and $48^4$ correct?

When should I use the power of product rule versus calculating first?

Does the power of product rule work with more than two factors?

What's the difference between this and the power of power rule?

Why can't I write $(6 \times 8)^4 = 6^4 \times 8$ ?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Calculate (5×7)³: Evaluating a Cubed Expression

Calculate (2×6)³: Evaluating the Cube of a Product

Calculate (9×7)⁴: Fourth Power of Product Exercise

Calculate (10×3)⁴: Fourth Power of a Product Expression

Calculate (6×8)⁴: Evaluating a Product Raised to Fourth Power

Exponent Rules with Product Operations

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why are both 64×84 6^4 \times 8^4 64×84 and 484 48^4 484 correct?

When should I use the power of product rule versus calculating first?

Does the power of product rule work with more than two factors?

What's the difference between this and the power of power rule?

Why can't I write (6×8)4=64×8 (6 \times 8)^4 = 6^4 \times 8 (6×8)4=64×8?

🌟 Unlock Your Math Potential

More Questions

Power of a Product

Calculate (5×7)³: Evaluating a Cubed Expression

Calculate (2×6)³: Evaluating the Cube of a Product

Calculate (9×7)⁴: Fourth Power of Product Exercise

Calculate (10×3)⁴: Fourth Power of a Product Expression

Why are both $6^4 \times 8^4$ and $48^4$ correct?

Why can't I write $(6 \times 8)^4 = 6^4 \times 8$ ?