Simplify 5^8 × 5^3: Multiplying Powers with Same Base

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

Because exponents represent repeated multiplication! $ 5^8 $ means 5 multiplied by itself 8 times, and $ 5^3 $ means 5 multiplied by itself 3 times. Together, that's 5 multiplied by itself 8 + 3 = 11 times.

Q: What if the bases are different?

This rule only works when the bases are the same! If you have different bases like $ 3^2 \times 5^4 $, you cannot combine them using this rule. The bases must be identical.

Q: When do I multiply exponents?

You multiply exponents when raising a power to another power, like $ (5^8)^3 = 5^{8 \times 3} = 5^{24} $. This is a completely different rule from multiplying powers!

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

Not necessarily! $ 5^{11} $ is often the simplest form and perfectly acceptable as your final answer. Only calculate the numerical value if specifically asked.

Q: What if I have three or more terms to multiply?

Same rule applies! Just keep adding all the exponents. For example: $ 5^2 \times 5^4 \times 5^6 = 5^{2+4+6} = 5^{12} $.

Exponent Laws with Same Base Multiplication

Simplify the following equation:

$5^8\times5^3=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's simplify this problem together.

00:11 According to exponent rules, when you multiply exponents with the same base, like A,

00:18 you keep the base and add the exponents, N plus M.

00:23 Now, let's apply this formula to our exercise.

00:27 We'll add the exponents and raise them to that power.

00:31 And there you have it, that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following equation:

$5^8\times5^3=$

Step-by-step solution

To simplify the expression $5^8 \times 5^3$ , we will use the exponent rule which states that when multiplying powers with the same base, we add the exponents:

Step-by-step:

Identify the base and exponents: Here, the base is $5$ , and the exponents are $8$ and $3$ .
Apply the multiplication of exponents rule: $5^8 \times 5^3 = 5^{8+3}$ .

Therefore, the correct answer is $5^{8+3}$ .

Final Answer

$5^{8+3}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add exponents
Technique: $5^8 \times 5^3 = 5^{8+3} = 5^{11}$
Check: Verify base stays same and exponents were added, not multiplied ✓

Common Mistakes

Avoid these frequent errors

Multiplying the exponents instead of adding
Don't multiply exponents like $5^8 \times 5^3 = 5^{24}$ ! This confuses the multiplication rule with the power rule. Always add exponents when multiplying powers with the same base.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we add the exponents instead of multiplying them?

Because exponents represent repeated multiplication! $5^8$ means 5 multiplied by itself 8 times, and $5^3$ means 5 multiplied by itself 3 times. Together, that's 5 multiplied by itself 8 + 3 = 11 times.

What if the bases are different?

This rule only works when the bases are the same! If you have different bases like $3^2 \times 5^4$ , you cannot combine them using this rule. The bases must be identical.

When do I multiply exponents?

You multiply exponents when raising a power to another power, like $(5^8)^3 = 5^{8 \times 3} = 5^{24}$ . This is a completely different rule from multiplying powers!

Do I need to calculate the final numerical answer?

Not necessarily! $5^{11}$ is often the simplest form and perfectly acceptable as your final answer. Only calculate the numerical value if specifically asked.

What if I have three or more terms to multiply?

Same rule applies! Just keep adding all the exponents. For example: $5^2 \times 5^4 \times 5^6 = 5^{2+4+6} = 5^{12}$ .

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