Simplify the Expression: 7⁵ × 2³ × 7² × 2⁴ Product of Powers

Q: Why can't I just multiply all the numbers together?

Because you have exponential expressions, not regular multiplication! The bases (7 and 2) have different exponent rules. You must group like bases and add their exponents separately.

Q: What if the bases were different, like 7 and 3?

If all bases are different, you cannot simplify further! The expression $ 7^5 \times 3^2 $ would stay as is because there are no like bases to combine.

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

Not unless asked! $ 7^7 \times 2^7 $ is the simplified form. Computing $ 7^7 = 823,543 $ and $ 2^7 = 128 $ would make it more complicated!

Q: Can I rearrange the terms before grouping?

Yes! Multiplication is commutative, so $ 7^5 \times 2^3 \times 7^2 \times 2^4 $ can be rearranged as $ 7^5 \times 7^2 \times 2^3 \times 2^4 $ to make grouping easier.

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

The product rule adds exponents when multiplying: $ a^m \times a^n = a^{m+n} $. The quotient rule subtracts exponents when dividing: $ a^m ÷ a^n = a^{m-n} $.

Product of Powers with Multiple Bases

Simplify the following equation:

$7^5\times2^3\times7^2\times2^4=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:03 We'll use the commutative law and arrange equal bases together

00:16 We'll use the formula for multiplying power by power

00:18 Any number (A) to the power of (M) multiplied by the same number (A) to the power of (N)

00:21 We'll get the same number (A) to the power of the sum of exponents (M+N)

00:24 We'll use this formula in our exercise

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

00:37 We'll keep the base and add the exponents

01:04 We'll use the same method for the second base

01:28 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:

$7^5\times2^3\times7^2\times2^4=$

Step-by-step solution

To solve this problem, we'll apply the laws of exponents to simplify the expression $7^5 \times 2^3 \times 7^2 \times 2^4$ .

Let's follow these steps:

Step 1: Identify like bases.
We have two like bases in the expression: 7 and 2.
Step 2: Apply the product of powers rule for each base separately.
For the base 7: $7^5 \times 7^2 = 7^{5+2} = 7^7$ .
For the base 2: $2^3 \times 2^4 = 2^{3+4} = 2^7$ .
Step 3: Combine the results.
The expression simplifies to $7^7 \times 2^7$ .

The simplified form of the original expression is therefore $7^7 \times 2^7$ .

Final Answer

$7^7\times2^7$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: Group like bases: $7^5 \times 7^2 = 7^{5+2} = 7^7$
Check: Count original exponents: 5+2=7 for base 7, 3+4=7 for base 2 ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents like 7^5 × 7^2 = 7^10! This gives a completely wrong answer because you're confusing power rules. Always add exponents when multiplying same bases: 7^5 × 7^2 = 7^(5+2) = 7^7.

Practice Quiz

Test your knowledge with interactive questions

$$ (2^3)^6 = $$

FAQ

Everything you need to know about this question

Why can't I just multiply all the numbers together?

Because you have exponential expressions, not regular multiplication! The bases (7 and 2) have different exponent rules. You must group like bases and add their exponents separately.

What if the bases were different, like 7 and 3?

If all bases are different, you cannot simplify further! The expression $7^5 \times 3^2$ would stay as is because there are no like bases to combine.

Do I need to calculate the final numerical answer?

Not unless asked! $7^7 \times 2^7$ is the simplified form. Computing $7^7 = 823,543$ and $2^7 = 128$ would make it more complicated!

Can I rearrange the terms before grouping?

Yes! Multiplication is commutative, so $7^5 \times 2^3 \times 7^2 \times 2^4$ can be rearranged as $7^5 \times 7^2 \times 2^3 \times 2^4$ to make grouping easier.

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

The product rule adds exponents when multiplying: $a^m \times a^n = a^{m+n}$ . The quotient rule subtracts exponents when dividing: $a^m ÷ a^n = a^{m-n}$ .

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