Simplify the Expression: 6²×6³×3² Using Exponent Rules

Q: Why can't I combine $ 6^5 $ and $ 3^2 $ ?

Because 6 and 3 are different bases! You can only use the rule $ a^m \times a^n = a^{m+n} $ when the bases are exactly the same. Think of it like adding apples and oranges - they stay separate.

Q: What if I want to calculate the actual number?

You could calculate $ 6^5 = 7776 $ and $ 3^2 = 9 $, then multiply to get 69,984. But the simplified form $ 6^5 \times 3^2 $ is usually the preferred answer.

Q: Can I rewrite 6 as $ 2 \times 3 $ to help combine terms?

Great thinking! Yes, $ 6^5 = (2 \times 3)^5 = 2^5 \times 3^5 $. Then you'd have $ 2^5 \times 3^5 \times 3^2 = 2^5 \times 3^7 $. Both forms are correct!

Q: How do I remember when to add exponents?

Use this simple check: "Same base? Add the exponents!" If the bases don't match exactly, keep them separate. $ 6^2 \times 6^3 $ becomes $ 6^5 $, but $ 6^5 \times 3^2 $ stays as is.

Q: What's the difference between this and $ (6 \times 3)^2 $ ?

Completely different! $ (6 \times 3)^2 = 18^2 = 324 $, while our expression $ 6^2 \times 6^3 \times 3^2 = 6^5 \times 3^2 = 69,984 $. Parentheses change everything!

Exponent Rules with Mixed Bases

Reduce the following equation:

$6^2\times6^3\times3^2=$

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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, the multiplication of exponents with the same base (A)

00:06 equals the same base raised to the sum of the exponents (N+M)

00:09 We'll apply this formula to our exercise, one operation at a time

00:13 We'll maintain the base and add up the exponents

00:21 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:

$6^2\times6^3\times3^2=$

Step-by-step solution

To simplify the expression $6^2 \times 6^3 \times 3^2$ , we will apply the rules for exponents:

Step 1: Identify powers with the same base. Here, we notice that $6^2$ and $6^3$ are powers with the base 6.
Step 2: Use the rule $a^m \times a^n = a^{m+n}$ to combine these powers: $6^2 \times 6^3 = 6^{2+3} = 6^5$ .
Step 3: The term $3^2$ remains unaffected by this operation because its base differs from that of 6. Therefore, it stays as $3^2$ .

Therefore, the simplified form of $6^2 \times 6^3 \times 3^2$ is $6^5 \times 3^2$ .

Final Answer

$6^5\times3^2$

Key Points to Remember

Essential concepts to master this topic

Same Base Rule: $a^m \times a^n = a^{m+n}$ only for identical bases
Technique: $6^2 \times 6^3 = 6^{2+3} = 6^5$ then keep $3^2$ separate
Check: Count exponents: 2+3=5 for base 6, base 3 stays as 2 ✓

Common Mistakes

Avoid these frequent errors

Adding all exponents together regardless of base
Don't add 2+3+2=7 to get $6^7$ or $18^7$ ! This ignores that 6 and 3 are different bases and creates completely wrong results. Always combine exponents only when the bases are identical.

Practice Quiz

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$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why can't I combine $6^5$ and $3^2$ ?

Because 6 and 3 are different bases! You can only use the rule $a^m \times a^n = a^{m+n}$ when the bases are exactly the same. Think of it like adding apples and oranges - they stay separate.

What if I want to calculate the actual number?

You could calculate $6^5 = 7776$ and $3^2 = 9$ , then multiply to get 69,984. But the simplified form $6^5 \times 3^2$ is usually the preferred answer.

Can I rewrite 6 as $2 \times 3$ to help combine terms?

Great thinking! Yes, $6^5 = (2 \times 3)^5 = 2^5 \times 3^5$ . Then you'd have $2^5 \times 3^5 \times 3^2 = 2^5 \times 3^7$ . Both forms are correct!

How do I remember when to add exponents?

Use this simple check: "Same base? Add the exponents!" If the bases don't match exactly, keep them separate. $6^2 \times 6^3$ becomes $6^5$ , but $6^5 \times 3^2$ stays as is.

What's the difference between this and $(6 \times 3)^2$ ?

Completely different! $(6 \times 3)^2 = 18^2 = 324$ , while our expression $6^2 \times 6^3 \times 3^2 = 6^5 \times 3^2 = 69,984$ . Parentheses change everything!

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Simplify the Expression: 6²×6³×3² Using Exponent Rules

Exponent Rules with Mixed Bases

❤️ 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 can't I combine $6^5$ and $3^2$ ?

What if I want to calculate the actual number?

Can I rewrite 6 as $2 \times 3$ to help combine terms?

How do I remember when to add exponents?

What's the difference between this and $(6 \times 3)^2$ ?

🌟 Unlock Your Math Potential

More Questions

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Simplify the Expression: 2³ × 3⁸ × 3⁹ × 2⁶

Simplify the Expression: a^-3 × y^7 × a^-4 × y^-5 Using Exponent Rules

Simplify a^(-5) × a^8 × x^3: Combining Multiple Exponents

Simplify a^(-3x) × a^b × a^b: Exponent Multiplication Practice

Simplify the Expression: 6²×6³×3² Using Exponent Rules

Exponent Rules with Mixed Bases

❤️ 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 can't I combine 65 6^5 65 and 32 3^2 32?

What if I want to calculate the actual number?

Can I rewrite 6 as 2×3 2 \times 3 2×3 to help combine terms?

How do I remember when to add exponents?

What's the difference between this and (6×3)2 (6 \times 3)^2 (6×3)2?

🌟 Unlock Your Math Potential

More Questions

Multiplication of Powers

Simplify the Expression: 2³ × 3⁸ × 3⁹ × 2⁶

Simplify the Expression: a^-3 × y^7 × a^-4 × y^-5 Using Exponent Rules

Simplify a^(-5) × a^8 × x^3: Combining Multiple Exponents

Simplify a^(-3x) × a^b × a^b: Exponent Multiplication Practice

Why can't I combine $6^5$ and $3^2$ ?

Can I rewrite 6 as $2 \times 3$ to help combine terms?

What's the difference between this and $(6 \times 3)^2$ ?