Reduce the Expression: 7⁶ × 13² × 4² × 8⁷ × 9² Using Laws of Exponents

Q: Why can I only combine terms with the same exponent?

The power of a product rule works backwards: $ (abc)^n = a^n \times b^n \times c^n $. If exponents don't match, like $ 7^6 \times 13^2 $, there's no way to write them as a single power!

Q: What if no terms have the same exponent?

That's okay! Your expression is already in simplest form. You can't reduce it further using exponent rules. Just make sure all bases are in their simplest form (like factoring composite numbers).

Q: Can I combine 8⁷ with any other terms in this problem?

No, because 8⁷ has a unique exponent. Only $ 13^2 \times 4^2 \times 9^2 $ can be combined since they all have exponent 2.

Q: How do I know which terms to group together?

Look for the same exponent! In this problem, scan for all terms with exponent 2: $ 13^2, 4^2, 9^2 $. These three can be grouped as $ (13 \times 4 \times 9)^2 $.

Q: Should I calculate the numbers inside the parentheses?

Usually no for this type of problem! The goal is to reduce the expression using exponent laws, not to find a numerical answer. Leave it as $ (13 \times 4 \times 9)^2 $.

Combining Exponents with Common Powers

Reduce the following equation:

$7^6\times13^2\times4^2\times8^7\times9^2=$

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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 laws of exponents, a product raised to the power (N)

00:07 equals the product where each factor is raised to the same power (N)

00:13 Identify all numbers whose exponents are equal

00:22 We will apply the formula to our exercise

00:30 Numbers with different exponents remain outside the parentheses

00:33 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:

$7^6\times13^2\times4^2\times8^7\times9^2=$

Step-by-step solution

Let's solve the problem by following these steps:

Step 1: Identify terms that share a common power. We have $13^2$ , $4^2$ , and $9^2$ , all raised to 2.

Step 2: Use the power of a product rule: $(a \times b \times c)^m = a^m \times b^m \times c^m$ .

Step 3: Combine these terms: $13^2 \times 4^2 \times 9^2 = (13 \times 4 \times 9)^2$ .

Step 4: Substitute back into the original expression:
$7^6 \times 13^2 \times 4^2 \times 8^7 \times 9^2 = 7^6 \times (13 \times 4 \times 9)^2 \times 8^7$ .

Therefore, the expression reduces to $7^6 \times (13 \times 4 \times 9)^2 \times 8^7$ .

Final Answer

$7^6\times\left(13\times4\times9\right)^2\times8^7$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Terms with same exponents can be combined using $(abc)^n = a^n \times b^n \times c^n$
Technique: Group terms: $13^2 \times 4^2 \times 9^2 = (13 \times 4 \times 9)^2$
Check: Verify by expanding back: $(13 \times 4 \times 9)^2 = 13^2 \times 4^2 \times 9^2$ ✓

Common Mistakes

Avoid these frequent errors

Trying to combine terms with different exponents
Don't group $7^6 \times 13^2$ as $(7 \times 13)^6$ = wrong exponents! You can only combine terms when they have the exact same power. Always identify matching exponents first, then group only those terms.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I only combine terms with the same exponent?

The power of a product rule works backwards: $(abc)^n = a^n \times b^n \times c^n$ . If exponents don't match, like $7^6 \times 13^2$ , there's no way to write them as a single power!

What if no terms have the same exponent?

That's okay! Your expression is already in simplest form. You can't reduce it further using exponent rules. Just make sure all bases are in their simplest form (like factoring composite numbers).

Can I combine 8⁷ with any other terms in this problem?

No, because 8⁷ has a unique exponent. Only $13^2 \times 4^2 \times 9^2$ can be combined since they all have exponent 2.

How do I know which terms to group together?

Look for the same exponent! In this problem, scan for all terms with exponent 2: $13^2, 4^2, 9^2$ . These three can be grouped as $(13 \times 4 \times 9)^2$ .

Should I calculate the numbers inside the parentheses?

Usually no for this type of problem! The goal is to reduce the expression using exponent laws, not to find a numerical answer. Leave it as $(13 \times 4 \times 9)^2$ .

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