Reduce the Expression: Product of a⁸ × b⁸ × c⁸

Q: Why can I combine these terms when the bases are different?

You can combine them because they have the same exponent (8)! The power of product rule works backwards: $ x^n \times y^n \times z^n = (xyz)^n $. Different bases with same exponents can be grouped.

Q: What's the difference between this and adding exponents?

Adding exponents only works with same bases: $ a^3 \times a^5 = a^8 $. Here we have different bases with same exponents, so we use the power of product rule instead.

Q: How do I know when to use parentheses?

Use parentheses when you're grouping multiple bases under one exponent. Without parentheses, $ abc^8 $ means only c is raised to the 8th power, but $ (abc)^8 $ means the entire product is raised to the 8th power.

Q: Can I leave the answer as the original expression?

While $ a^8 \times b^8 \times c^8 $ is mathematically correct, $ (abc)^8 $ is the simplified form. Math questions asking to "reduce" want the most compact version.

Q: What if the exponents were different numbers?

If exponents don't match, you cannot use this rule! For example, $ a^5 \times b^3 \times c^8 $ stays as is because there's no common exponent to factor out.

Exponent Rules with Product Simplification

Reduce the following equation:

$a^8\times b^8\times c^8=$

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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 exponent laws, a product raised to the power (N)

00:07 equals the product of each factor raised to the power (N)

00:12 We will apply this formula to our exercise

00:20 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Reduce the following equation:

$a^8\times b^8\times c^8=$

Step-by-step solution

To reduce the expression $a^8 \times b^8 \times c^8$ , we can apply the Power of a Product Rule, which states that when multiplying powers with the same exponent across different bases, we can combine them into a single power. Specifically, this rule is written as:

$(x^m \times y^m \times z^m) = (x \times y \times z)^m.$

Applying this rule to our expression $a^8 \times b^8 \times c^8$ , we identify $x = a$ , $y = b$ , and $z = c$ , all with the exponent $m = 8$ . Therefore, we can simplify the expression to:

$(a \times b \times c)^8.$

Thus, the reduced form of the given expression is:

$(a \times b \times c)^8$ .

Final Answer

$\left(a\times b\times c\right)^8$

Key Points to Remember

Essential concepts to master this topic

Rule: When bases differ but exponents match, use power of product
Technique: Convert $a^8 \times b^8 \times c^8$ to $(a \times b \times c)^8$
Check: Expand back: $(abc)^8 = a^8b^8c^8$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of using power of product rule
Don't add the exponents to get $a^{24} \times b^{24} \times c^{24}$ = completely wrong expression! This confuses the product rule with the power rule. Always group same exponents using $(xyz)^n = x^n \times y^n \times z^n$ in reverse.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I combine these terms when the bases are different?

You can combine them because they have the same exponent (8)! The power of product rule works backwards: $x^n \times y^n \times z^n = (xyz)^n$ . Different bases with same exponents can be grouped.

What's the difference between this and adding exponents?

Adding exponents only works with same bases: $a^3 \times a^5 = a^8$ . Here we have different bases with same exponents, so we use the power of product rule instead.

How do I know when to use parentheses?

Use parentheses when you're grouping multiple bases under one exponent. Without parentheses, $abc^8$ means only c is raised to the 8th power, but $(abc)^8$ means the entire product is raised to the 8th power.

Can I leave the answer as the original expression?

While $a^8 \times b^8 \times c^8$ is mathematically correct, $(abc)^8$ is the simplified form. Math questions asking to "reduce" want the most compact version.

What if the exponents were different numbers?

If exponents don't match, you cannot use this rule! For example, $a^5 \times b^3 \times c^8$ stays as is because there's no common exponent to factor out.

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