Simplify 4²×3⁵×4³×3²: Product of Powers Problem

Q: Why can't I just add all the exponents together?

The product rule $ a^m \times a^n = a^{m+n} $ only works when the bases are identical. Since 4 and 3 are different bases, you must handle them separately!

Q: What if I have more than two of the same base?

No problem! Just keep adding the exponents. For example: $ 2^3 \times 2^4 \times 2^1 = 2^{3+4+1} = 2^8 $. The rule works for any number of terms with the same base.

Q: Do I need to calculate the final numerical answer?

Usually no! Leaving your answer as $ 4^5 \times 3^7 $ is the simplified form. Only calculate the numerical value if specifically asked to do so.

Q: What if the bases can be rewritten with a common base?

Great question! Sometimes you can rewrite bases like $ 8^2 \times 2^3 $ as $ (2^3)^2 \times 2^3 = 2^6 \times 2^3 = 2^9 $. But in this problem, 4 and 3 don't share a simple common base.

Q: Can I rearrange the terms before grouping?

Absolutely! Multiplication is commutative, so $ 4^2 \times 3^5 \times 4^3 \times 3^2 $ can be rearranged as $ 4^2 \times 4^3 \times 3^5 \times 3^2 $ to make grouping easier.

Exponent Laws with Multiple Bases

Simplify the following equation:

$4^2\times3^5\times4^3\times3^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:05 Let's use the substitution law and arrange the equal bases together

00:13 Any number (A) to the power of (M) multiplied by the same number (A) to the power of (N)

00:16 We get the same number (A) to the power of the sum of exponents (M+N)

00:19 Let's use this formula in our exercise

00:22 And let's equate the numbers with the variables in the formula

00:32 Let's keep the base and sum the exponents

00:53 Let's use the same method for the second base

01:16 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:

$4^2\times3^5\times4^3\times3^2=$

Step-by-step solution

To simplify the given expression $4^2 \times 3^5 \times 4^3 \times 3^2$ , we will follow these steps:

Step 1: Identify and group similar bases.
Step 2: Apply the rule for multiplying like bases.
Step 3: Simplify the expression.

Now, let's go through each step thoroughly:

Step 1: Identify and group similar bases:
We see two distinct bases here: 4 and 3.

Step 2: Apply the rule for multiplying like bases:
For base 4: Combine $4^2$ and $4^3$ , using the rule $a^m \times a^n = a^{m+n}$ .

Add the exponents for base 4: $2 + 3 = 5$ , thus, $4^2 \times 4^3 = 4^5$ .

For base 3: Combine $3^5$ and $3^2$ , still using the same exponent rule.

Add the exponents for base 3: $5 + 2 = 7$ , resulting in $3^5 \times 3^2 = 3^7$ .

Step 3: Simplify the expression:
The simplified expression is $4^5 \times 3^7$ .

Therefore, the final simplified expression is $4^5 \times 3^7$ .

Final Answer

$4^5\times3^7$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Grouping: Collect like bases: $4^2 \times 4^3 = 4^{2+3} = 4^5$
Verify: Check that all like bases are combined: $4^5 \times 3^7$ has no repeated bases ✓

Common Mistakes

Avoid these frequent errors

Adding exponents across different bases
Don't add exponents like 2+5+3+2=12 to get $4^{12}$ or $3^{12}$ ! This ignores that 4 and 3 are different bases. Always group same bases first: $4^2 \times 4^3 = 4^5$ and $3^5 \times 3^2 = 3^7$ separately.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just add all the exponents together?

The product rule $a^m \times a^n = a^{m+n}$ only works when the bases are identical. Since 4 and 3 are different bases, you must handle them separately!

What if I have more than two of the same base?

No problem! Just keep adding the exponents. For example: $2^3 \times 2^4 \times 2^1 = 2^{3+4+1} = 2^8$ . The rule works for any number of terms with the same base.

Do I need to calculate the final numerical answer?

Usually no! Leaving your answer as $4^5 \times 3^7$ is the simplified form. Only calculate the numerical value if specifically asked to do so.

What if the bases can be rewritten with a common base?

Great question! Sometimes you can rewrite bases like $8^2 \times 2^3$ as $(2^3)^2 \times 2^3 = 2^6 \times 2^3 = 2^9$ . But in this problem, 4 and 3 don't share a simple common base.

Can I rearrange the terms before grouping?

Absolutely! Multiplication is commutative, so $4^2 \times 3^5 \times 4^3 \times 3^2$ can be rearranged as $4^2 \times 4^3 \times 3^5 \times 3^2$ to make grouping easier.

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