Complete the Expression: Writing (ax×3)^b in Standard Form

Q: Why does the exponent apply to all three factors?

The Power of a Product Rule states that when you raise a product to an exponent, every factor inside the parentheses gets that exponent. Think of it as distributing the exponent to each term!

Q: What if one of the factors is already raised to a power?

Use the Power of a Power Rule! If you have $ (a^2 \times x \times 3)^b $, then $ a^2 $ becomes $ a^{2b} $ by multiplying the exponents.

Q: How do I remember when to use this rule?

Look for parentheses with an exponent outside! Whenever you see $ (\text{stuff})^{\text{exponent}} $, that's your cue to apply the power rule to everything inside the parentheses.

Q: What's the difference between this and distributing addition?

Big difference! With multiplication like $ (a \times x \times 3)^b $, the exponent applies to each factor. With addition like $ (a + x + 3)^b $, you can't just distribute—you need to use binomial expansion!

Exponent Rules with Product Expressions

Insert the corresponding expression:

$\left(a\times x\times3\right)^b=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this math problem together.

00:11 Remember, with exponents, if you're multiplying terms, each raised to the power of N,

00:16 you raise each factor to the power of N, one at a time.

00:21 Let's use this rule in our exercise now.

00:24 And there you go, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\left(a\times x\times3\right)^b=$

Step-by-step solution

To solve this problem, we need to apply the Power of a Product Rule, which states that if you raise a product to an exponent, you raise each factor in the product to that exponent.

We start with the expression $(a \times x \times 3)^b$ . By applying the Power of a Product Rule, we can rewrite it as:

Raise $a$ to the power of $b$ , giving us $a^b$ .
Raise $x$ to the power of $b$ , giving us $x^b$ .
Raise $3$ to the power of $b$ , giving us $3^b$ .

Therefore, the expression $(a \times x \times 3)^b$ becomes $a^b \times x^b \times 3^b$ . This matches choice 3 from the provided options.

This demonstrates the proper application of the Power of a Product Rule. Thus, the expression is simplified to:

$a^b \times x^b \times 3^b$ .

Final Answer

$a^b\times x^b\times3^b$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: Each factor gets raised to the outer exponent
Technique: Apply exponent b to each term: $a^b \times x^b \times 3^b$
Check: Verify by expanding: $(2 \times x \times 3)^2 = 4 \times x^2 \times 9$ ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to the first factor
Don't just write $a^b \times x \times 3$ = incomplete result! This ignores the power rule and leaves factors unchanged. Always apply the outer exponent to every single factor inside the parentheses.

Practice Quiz

Test your knowledge with interactive questions

$$ (4^2)^3+(g^3)^4= $$

FAQ

Everything you need to know about this question

Why does the exponent apply to all three factors?

The Power of a Product Rule states that when you raise a product to an exponent, every factor inside the parentheses gets that exponent. Think of it as distributing the exponent to each term!

What if one of the factors is already raised to a power?

Use the Power of a Power Rule! If you have $(a^2 \times x \times 3)^b$ , then $a^2$ becomes $a^{2b}$ by multiplying the exponents.

Can I simplify the expression further?

Yes! You can rearrange the terms in any order since multiplication is commutative. For example, $a^b \times x^b \times 3^b$ could also be written as $3^b \times a^b \times x^b$ .

How do I remember when to use this rule?

Look for parentheses with an exponent outside! Whenever you see $(\text{stuff})^{\text{exponent}}$ , that's your cue to apply the power rule to everything inside the parentheses.

What's the difference between this and distributing addition?

Big difference! With multiplication like $(a \times x \times 3)^b$ , the exponent applies to each factor. With addition like $(a + x + 3)^b$ , you can't just distribute—you need to use binomial expansion!

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