Simplify 6^5 × 6^7: Exponential Multiplication Practice

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

Exponents show repeated multiplication! $ 6^5 = 6 \times 6 \times 6 \times 6 \times 6 $ (5 sixes) and $ 6^7 $ means 7 more sixes. Combined, that's 5 + 7 = 12 sixes multiplied together.

Q: What if the bases are different, like 6² × 7³?

You cannot use this rule when bases are different! $ 6^2 \times 7^3 $ stays as is because 6 and 7 are different numbers. The exponent addition rule only works with identical bases.

Q: Does this work with variables too?

Absolutely! $ x^3 \times x^4 = x^{3+4} = x^7 $. The same rule applies whether you're working with numbers or variables - just make sure the bases are identical.

Q: How can I remember this rule easily?

Think of it as collecting factors: if you have 5 copies of 6 multiplied by 7 more copies of 6, you end up with 5 + 7 = 12 total copies of 6 being multiplied!

Q: What happens if I add the bases instead?

That's completely wrong! $ 6^5 \times 6^7 \neq (6+6)^{5+7} $. The bases stay the same - only the exponents get added when multiplying powers with identical bases.

Exponent Laws with Same Base Multiplication

Simplify the following equation:

$6^5\times6^7=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this math problem together.

00:10 When multiplying numbers with the same base, A, you add their exponents.

00:15 So, A raised to the power of N plus M.

00:19 Let's use this rule to solve our exercise.

00:23 Add up the exponents and raise A to this power.

00:27 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:

$6^5\times6^7=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given expression
Step 2: Apply the exponent rule for multiplication of powers with the same base
Step 3: Simplify the expression by adding the exponents

Now, let's work through each step:
Step 1: The given expression is $6^5 \times 6^7$ . Here, the base is 6, and the exponents are 5 and 7.
Step 2: We apply the exponent rule, which states that when multiplying two powers with the same base, we add the exponents. Therefore, we have:

$6^5 \times 6^7 = 6^{5+7}$

Step 3: Add the exponents: $5 + 7 = 12$ . Thus, the expression simplifies to:

$6^{12}$

Therefore, the solution to the problem is $6^{12}$ .

Final Answer

$6^{12}$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: $6^5 \times 6^7 = 6^{5+7} = 6^{12}$
Check: Count total factors: 6⁵ has 5 sixes, 6⁷ has 7 sixes = 12 sixes total ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of adding them
Don't multiply the exponents 5 × 7 = 35 to get $6^{35}$ ! This creates an astronomically large number that doesn't match the original problem. 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 exponents instead of multiplying them?

Exponents show repeated multiplication! $6^5 = 6 \times 6 \times 6 \times 6 \times 6$ (5 sixes) and $6^7$ means 7 more sixes. Combined, that's 5 + 7 = 12 sixes multiplied together.

What if the bases are different, like 6² × 7³?

You cannot use this rule when bases are different! $6^2 \times 7^3$ stays as is because 6 and 7 are different numbers. The exponent addition rule only works with identical bases.

Does this work with variables too?

Absolutely! $x^3 \times x^4 = x^{3+4} = x^7$ . The same rule applies whether you're working with numbers or variables - just make sure the bases are identical.

How can I remember this rule easily?

Think of it as collecting factors: if you have 5 copies of 6 multiplied by 7 more copies of 6, you end up with 5 + 7 = 12 total copies of 6 being multiplied!

What happens if I add the bases instead?

That's completely wrong! $6^5 \times 6^7 \neq (6+6)^{5+7}$ . The bases stay the same - only the exponents get added when multiplying powers with identical bases.

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