Simplify the Expression: 5³ × 2⁴ × 5² × 2³ Using Laws of Exponents

Q: Why do I add the exponents instead of multiplying them?

The product rule says $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 5^3 = 5 \times 5 \times 5 $ and $ 5^2 = 5 \times 5 $, so together you have five 5's multiplied = $ 5^5 $!

Q: What if the bases are different like 5³ × 2⁴?

You cannot combine different bases using exponent rules! Keep them separate: $ 5^3 \times 2^4 $ stays as is. Only combine terms with the same base.

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

Usually no! The simplified form $ 5^5 \times 2^7 $ is the correct answer unless specifically asked to evaluate. This form clearly shows the exponent rules were applied correctly.

Q: How do I remember which exponent rule to use?

Multiplication = Addition: When you see multiplication of like bases, add the exponents. Power of a power = Multiplication: When you see $ (a^m)^n $, multiply the exponents.

Q: Can I rearrange the terms before combining?

Absolutely! Multiplication is commutative, so you can rearrange: $ 5^3 \times 2^4 \times 5^2 \times 2^3 = 5^3 \times 5^2 \times 2^4 \times 2^3 $ to group like bases together.

Exponent Laws with Like Base Terms

Simplify the following equation:

$5^3\times2^4\times5^2\times2^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's begin!

00:13 We'll use the commutative law to group the same bases together.

00:25 Now, we'll apply the formula for multiplying powers.

00:29 If we have A to the power of M, times A to the power of N...

00:35 It equals A to the power of M plus N, the sum of the exponents.

00:40 We'll use this formula in our exercise now.

00:43 Let's substitute the numbers with unknowns into this formula.

00:47 Keep the base the same and just add the exponents together.

01:07 We'll do the same process for the second base.

01:30 And there you have it, that's 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:

$5^3\times2^4\times5^2\times2^3=$

Step-by-step solution

Let's simplify the expression $5^3 \times 2^4 \times 5^2 \times 2^3$ using the rules for exponents. We'll apply the product of powers rule, which states that when multiplying like bases, you can add the exponents.

Step 1: Focus on terms with the same base.
Combine $5^3$ and $5^2$ . Since both terms have the base $5$ , we apply the rule $a^m \times a^n = a^{m+n}$ : $5^3 \times 5^2 = 5^{3+2} = 5^5$
Step 2: Combine $2^4$ and $2^3$ . Similarly, for the base $2$ : $2^4 \times 2^3 = 2^{4+3} = 2^7$

After simplification, the expression becomes:
$5^5 \times 2^7$

Final Answer

$5^5\times2^7$

Key Points to Remember

Essential concepts to master this topic

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

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply exponents like 5³ × 5² = 5⁶ = wrong answer! This confuses the product rule with the power rule. Always add exponents when multiplying like bases: 5³ × 5² = 5³⁺² = 5⁵.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I add the exponents instead of multiplying them?

The product rule says $a^m \times a^n = a^{m+n}$ . Think of it this way: $5^3 = 5 \times 5 \times 5$ and $5^2 = 5 \times 5$ , so together you have five 5's multiplied = $5^5$ !

What if the bases are different like 5³ × 2⁴?

You cannot combine different bases using exponent rules! Keep them separate: $5^3 \times 2^4$ stays as is. Only combine terms with the same base.

Do I need to calculate the final numerical answer?

Usually no! The simplified form $5^5 \times 2^7$ is the correct answer unless specifically asked to evaluate. This form clearly shows the exponent rules were applied correctly.

How do I remember which exponent rule to use?

Multiplication = Addition: When you see multiplication of like bases, add the exponents. Power of a power = Multiplication: When you see $(a^m)^n$ , multiply the exponents.

Can I rearrange the terms before combining?

Absolutely! Multiplication is commutative, so you can rearrange: $5^3 \times 2^4 \times 5^2 \times 2^3 = 5^3 \times 5^2 \times 2^4 \times 2^3$ to group like bases together.

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