Simplify the Expression: a⁴ × b⁴ Product of Powers

Q: Why can't I add the exponents 4 + 4 = 8?

You can only add exponents when multiplying powers with the same base. Since $ a^4 $ and $ b^4 $ have different bases (a and b), you must use the power of product rule instead.

Q: How is this different from something like a⁴ × a⁴?

Great question! With $ a^4 \times a^4 $, both terms have the same base (a), so you add exponents: $ a^{4+4} = a^8 $. But $ a^4 \times b^4 $ has different bases, so you combine them as $ (a \times b)^4 $.

Q: Can I write (ab)⁴ instead of (a × b)⁴?

Absolutely! $ (ab)^4 $ and $ (a \times b)^4 $ mean exactly the same thing. In algebra, we often omit the multiplication symbol between variables for cleaner notation.

Q: What if the exponents were different, like a³ × b⁴?

If the exponents are different, you cannot combine them using the power of product rule. The expression $ a^3 \times b^4 $ is already in its simplest form and cannot be simplified further.

Q: How do I remember which exponent rule to use?

Look at the bases first! If bases are the same, add exponents. If bases are different but exponents are the same, use power of product rule. If both bases and exponents are different, the expression is already simplified.

Exponent Rules with Product Power

Insert the corresponding expression:

$a^4\times b^4=$

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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.

00:11 When you multiply numbers, each raised to a power, N, you can rewrite it as all numbers in parentheses, then raise them to power, N.

00:21 We'll use this method in our exercise now.

00:24 Great work, let's see how this formula solves our problem.

00:28 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$a^4\times b^4=$

Step-by-step solution

To solve the problem, we need to simplify the expression $a^4 \times b^4$ to a single power using exponent rules.

Here’s a step-by-step explanation:

Step 1: Identify the expression that needs simplification: $a^4 \times b^4$ .
Step 2: Apply the exponent rule called the power of a product: $(x \times y)^n = x^n \times y^n$ .
Step 3: We can reverse this property. If we have $x^n \times y^n$ , we can combine it as $(x \times y)^n$ .
Step 4: Apply this to our expression: $a^4 \times b^4 = (a \times b)^4$ . Using the power of a product, $a^4 \times b^4 = (a \times b)^4$ .

Thus, the expression $a^4 \times b^4$ simplifies to $(a \times b)^4$ .

Final Answer

$\left(a\times b\right)^4$

Key Points to Remember

Essential concepts to master this topic

Power of Product Rule: When multiplying powers with same exponent, combine bases
Technique: Reverse the rule: $a^4 \times b^4 = (a \times b)^4$
Check: Expand back: $(a \times b)^4 = a^4 \times b^4$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents when bases are different
Don't treat $a^4 \times b^4$ like $a^{4+4} = a^8$ ! Adding exponents only works when the bases are identical. Always check if bases match before applying any exponent rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I add the exponents 4 + 4 = 8?

You can only add exponents when multiplying powers with the same base. Since $a^4$ and $b^4$ have different bases (a and b), you must use the power of product rule instead.

How is this different from something like a⁴ × a⁴?

Great question! With $a^4 \times a^4$ , both terms have the same base (a), so you add exponents: $a^{4+4} = a^8$ . But $a^4 \times b^4$ has different bases, so you combine them as $(a \times b)^4$ .

Can I write (ab)⁴ instead of (a × b)⁴?

Absolutely! $(ab)^4$ and $(a \times b)^4$ mean exactly the same thing. In algebra, we often omit the multiplication symbol between variables for cleaner notation.

What if the exponents were different, like a³ × b⁴?

If the exponents are different, you cannot combine them using the power of product rule. The expression $a^3 \times b^4$ is already in its simplest form and cannot be simplified further.

How do I remember which exponent rule to use?

Look at the bases first! If bases are the same, add exponents. If bases are different but exponents are the same, use power of product rule. If both bases and exponents are different, the expression is already simplified.

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