Simplify the Expression: 2^10 × 3^6 × 2^5 × 3^2 Using Power Rules

Q: Why do I group terms with the same base together?

The product rule only works when bases are identical! You can't combine $ 2^{10} \times 3^6 $ directly, but you can combine $ 2^{10} \times 2^5 $ into $ 2^{15} $.

Q: Can I multiply the bases together to get one base?

No! Don't turn $ 2^{15} \times 3^8 $ into $ 6^{something} $. The product rule only applies when bases are already the same, not when you make them the same.

Q: How do I remember to add exponents and not multiply them?

Think of exponents as repeated multiplication. $ 2^{10} \times 2^5 $ means you're multiplying 2 ten times, then five more times, for a total of 15 times!

Q: What's the difference between this and the power rule?

The product rule is for multiplying powers with the same base (add exponents). The power rule is for raising a power to another power (multiply exponents). This problem uses only the product rule.

Exponent Rules with Multiple Bases

Simplify the following equation:

$2^{10}\times3^6\times2^5\times3^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:06 We'll use the commutative law and arrange equal bases together

00:15 We'll use the formula for multiplying power by power

00:17 Any number (A) to the power of (M) multiplied by the same number (A) to the power of (N)

00:20 We get the same number (A) to the power of the sum of exponents (M+N)

00:23 We'll use this formula in our exercise

00:27 And we'll equate the numbers to the variables in the formula

00:35 We'll keep the base and combine the exponents

01:06 We'll calculate the sums

01:11 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$2^{10}\times3^6\times2^5\times3^2=$

Step-by-step solution

To solve this problem, we'll simplify the expression $2^{10} \times 3^6 \times 2^5 \times 3^2$ using the rules of exponents. Here are the steps:

Step 1: Apply the product of powers property to the base 2 terms. The expression $2^{10} \times 2^5$ simplifies to:
$2^{10+5} = 2^{15}$
Step 2: Apply the product of powers property to the base 3 terms. The expression $3^6 \times 3^2$ simplifies to:
$3^{6+2} = 3^8$
Step 3: Combine the simplified terms to form the complete simplified expression:
$2^{15} \times 3^8$

Therefore, the simplified form of the equation is $2^{15} \times 3^8$ .

Final Answer

$2^{15}\times3^8$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: Group like bases: $2^{10} \times 2^5 = 2^{15}$
Check: Count exponents carefully: 10+5=15 and 6+2=8 gives $2^{15} \times 3^8$ ✓

Common Mistakes

Avoid these frequent errors

Adding coefficients instead of exponents
Don't add the base numbers like 2+2=4 or multiply exponents like 10×5=50! This creates completely wrong expressions like $4^{50}$ . Always keep the base unchanged and add only the exponents when using the product rule.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I group terms with the same base together?

The product rule only works when bases are identical! You can't combine $2^{10} \times 3^6$ directly, but you can combine $2^{10} \times 2^5$ into $2^{15}$ .

What if the bases are different like 2 and 3?

Keep them separate! Different bases like 2 and 3 cannot be combined using exponent rules. Your final answer will have both bases: $2^{15} \times 3^8$ .

Can I multiply the bases together to get one base?

No! Don't turn $2^{15} \times 3^8$ into $6^{something}$ . The product rule only applies when bases are already the same, not when you make them the same.

How do I remember to add exponents and not multiply them?

Think of exponents as repeated multiplication. $2^{10} \times 2^5$ means you're multiplying 2 ten times, then five more times, for a total of 15 times!

What's the difference between this and the power rule?

The product rule is for multiplying powers with the same base (add exponents). The power rule is for raising a power to another power (multiply exponents). This problem uses only the product rule.

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