Simplify the Expression: a²×a⁵×a³ Using Exponent Rules

Q: What if the bases are different letters?

You cannot combine them! The product rule only works with identical bases. So $ a^2 \times b^3 $ stays as $ a^2b^3 $.

Q: How is this different from the power of a power rule?

Power of a power uses multiplication: $ (a^2)^3 = a^{2 \times 3} = a^6 $. Product of powers uses addition: $ a^2 \times a^3 = a^{2+3} = a^5 $.

Q: Can I use this rule with more than three terms?

Absolutely! Just keep adding all the exponents. For example: $ a^1 \times a^4 \times a^2 \times a^3 = a^{1+4+2+3} = a^{10} $.

Q: What if one of the exponents is 1?

Remember that $ a^1 = a $, so just add 1 to your sum! For instance: $ a \times a^5 = a^1 \times a^5 = a^{1+5} = a^6 $.

Product of Powers with Same Base

Reduce the following equation:

$a^2\times a^5\times a^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 We'll use the formula for multiplying power by power

00:04 Any number (A) to the power of (M) times the same number (A) to the power of (N)

00:07 We'll get the same number (A) to the power of the sum of exponents (M+N)

00:10 We'll use this formula in our exercise

00:13 And we'll equate the numbers with the variables in the formula

00:32 We'll keep the base and sum the exponents

00:54 We'll use the same method again

01:07 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

Reduce the following equation:

$a^2\times a^5\times a^3=$

Step-by-step solution

To reduce the expression $a^2 \times a^5 \times a^3$ , we will apply the product of powers property of exponents. This property states that when multiplying expressions with the same base, we add their exponents.

Step 1: Identify the exponents.
The expression involves the same base $a$ with exponents: 2, 5, and 3.
Step 2: Add the exponents.
According to the product of powers property, $a^2 \times a^5 \times a^3 = a^{2+5+3}$ .
Step 3: Simplify the expression.
Calculate the sum of the exponents: $2 + 5 + 3 = 10$ . Therefore, the expression simplifies to $a^{10}$ .

Ultimately, the solution to the problem is $a^{10}$ . Among the provided choices, is correct: $a^{10}$ . The other options $a^5$ , $a^8$ , and $a^4$ do not correctly reflect the sum of the exponents as calculated.

Final Answer

$a^{10}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: $a^2 \times a^5 \times a^3 = a^{2+5+3} = a^{10}$
Check: Count total factors: a·a·a·a·a·a·a·a·a·a = 10 factors ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't multiply 2 × 5 × 3 = 30 to get $a^{30}$ ! This confuses the product rule with the power of a power rule. Always add exponents when multiplying same bases: 2 + 5 + 3 = 10.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add exponents when multiplying?

Because exponents count how many times you multiply the base! $a^2 \times a^3$ means (a·a) × (a·a·a), which gives you 5 total factors of a, so $a^5$ .

What if the bases are different letters?

You cannot combine them! The product rule only works with identical bases. So $a^2 \times b^3$ stays as $a^2b^3$ .

How is this different from the power of a power rule?

Power of a power uses multiplication: $(a^2)^3 = a^{2 \times 3} = a^6$ . Product of powers uses addition: $a^2 \times a^3 = a^{2+3} = a^5$ .

Can I use this rule with more than three terms?

Absolutely! Just keep adding all the exponents. For example: $a^1 \times a^4 \times a^2 \times a^3 = a^{1+4+2+3} = a^{10}$ .

What if one of the exponents is 1?

Remember that $a^1 = a$ , so just add 1 to your sum! For instance: $a \times a^5 = a^1 \times a^5 = a^{1+5} = a^6$ .

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