Simplify the Expression: 5³ × 5⁶ × 5² Using Laws of Exponents

Q: Why do we add exponents instead of multiplying them?

The product rule says $ a^m \times a^n = a^{m+n} $. Think of it this way: $ 5^3 $ means 5×5×5, so when you multiply by $ 5^6 $, you're adding 6 more 5's to your multiplication!

Q: What happens if the bases are different?

The product rule only works with the same base. For example, $ 3^2 \times 5^4 $ cannot be simplified using exponent laws - you'd have to calculate each power separately first.

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

Absolutely! You can multiply as many same-base powers as you want. Just add all the exponents together. For example: $ 2^1 \times 2^3 \times 2^5 \times 2^2 = 2^{1+3+5+2} = 2^{11} $

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

Usually no! In most cases, leaving your answer as $ 5^{11} $ is perfectly acceptable and often preferred. Only calculate the numerical value if specifically asked.

Exponent Laws with Same Base Multiplication

Simplify the following equation:

$5^3\times5^6\times5^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the laws of exponents, the multiplication of exponents with an equal base (A)

00:07 equals the same base raised to the sum of the exponents (N+M)

00:11 We will apply this formula to our exercise

00:14 We'll maintain the base and add up the exponents

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$5^3\times5^6\times5^2=$

Step-by-step solution

To solve the problem of simplifying the expression $5^3 \times 5^6 \times 5^2$ , follow these steps:

Step 1: Understand that the expression involves multiplying powers with the same base.
Step 2: Apply the formula for multiplying powers: $a^m \times a^n = a^{m+n}$ .
Step 3: Combine the exponents by adding them together.

Now, let's work through these steps in detail:
Step 1: Recognize the base is 5, with exponents 3, 6, and 2.
Step 2: Since all terms have the base 5, use the formula for multiplying powers, resulting in a single term where the exponents are added: $5^{3+6+2}$ .
Step 3: Calculate the sum of the exponents: $3 + 6 + 2 = 11$ .

Hence, the correct answer is $5^{3+6+2}$ which simplifies to $5^{11}$ .

Final Answer

$5^{3+6+2}$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add the exponents together
Technique: $5^3 \times 5^6 \times 5^2 = 5^{3+6+2} = 5^{11}$
Check: Base stays the same, only exponents combine: 5 base with 11 total power ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding them
Don't calculate $5^{3\times6\times2} = 5^{36}$ when multiplying powers! This gives a completely wrong answer because you're using the wrong operation. Always add exponents when multiplying same bases: $5^{3+6+2} = 5^{11}$ .

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 instead of multiplying them?

The product rule says $a^m \times a^n = a^{m+n}$ . Think of it this way: $5^3$ means 5×5×5, so when you multiply by $5^6$ , you're adding 6 more 5's to your multiplication!

What happens if the bases are different?

The product rule only works with the same base. For example, $3^2 \times 5^4$ cannot be simplified using exponent laws - you'd have to calculate each power separately first.

Can I use this rule with more than three terms?

Absolutely! You can multiply as many same-base powers as you want. Just add all the exponents together. For example: $2^1 \times 2^3 \times 2^5 \times 2^2 = 2^{1+3+5+2} = 2^{11}$

Do I need to calculate the final numerical answer?

Usually no! In most cases, leaving your answer as $5^{11}$ is perfectly acceptable and often preferred. Only calculate the numerical value if specifically asked.

What if one of the exponents is negative?

The same rule applies! Just add the exponents, including negative ones. For example: $4^5 \times 4^{-2} = 4^{5+(-2)} = 4^3$

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