Solve for n in the Equation 6^n = 6×6×6: Exponent Value Problem

Q: How do I know what the exponent should be?

Simply count how many times the base number appears in the multiplication. In $ 6\cdot6\cdot6 $, the number 6 appears 3 times, so the exponent is 3.

Q: What's the difference between 6×3 and 6³?

$ 6 \times 3 = 18 $ (multiplication), but $ 6^3 = 6\cdot6\cdot6 = 216 $ (repeated multiplication). The exponent tells you how many times to multiply the base by itself.

Q: Can I just multiply 6×6×6 to check my answer?

Absolutely! $ 6\times6\times6 = 216 $, and $ 6^3 = 216 $ too. If both expressions give the same result, your exponent is correct.

Q: What if there were more 6's in the multiplication?

Just keep counting! If you had $ 6\cdot6\cdot6\cdot6\cdot6 $, that would be $ 6^5 $ because the base 6 appears 5 times.

Q: Why can't I just look at the answer 216 to find n?

While you could work backwards, it's much easier to count the factors directly. Plus, this method works even when the numbers get really big and hard to calculate!

Exponent Notation with Repeated Multiplication

Find the value of n:

$6^n=6\cdot6\cdot6$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find the right exponent.

00:11 When any number, like A, multiplies itself, we call it squared.

00:16 When A multiplies itself N times, it's A to the power of N.

00:22 Let's apply this idea to our exercise.

00:25 We'll count how many times 6 multiplies itself.

00:30 This gives us the correct exponent.

00:34 And there you have it; that's the answer.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the value of n:

$6^n=6\cdot6\cdot6$ ?

Step-by-step solution

We use the formula: $a\times a=a^2$

In the formula, we see that the power shows the number of terms that are multiplied, that is, two times

Since in the exercise we multiply 6 three times, it means that we have 3 terms.

Therefore, the power, which is n in this case, will be 3.

Final Answer

$n=3$

Key Points to Remember

Essential concepts to master this topic

Rule: Exponent equals the number of times the base is multiplied
Technique: Count multiplication factors: $6\cdot6\cdot6$ has 3 factors
Check: Verify $6^3 = 6\cdot6\cdot6 = 216$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of counting multiplication factors
Don't add the bases (6+6+6=18) or confuse the number of factors with the result! This completely misses the concept of exponents. Always count how many times the base appears in the multiplication.

Practice Quiz

Test your knowledge with interactive questions

$$ 11^2= $$

FAQ

Everything you need to know about this question

How do I know what the exponent should be?

Simply count how many times the base number appears in the multiplication. In $6\cdot6\cdot6$ , the number 6 appears 3 times, so the exponent is 3.

What's the difference between 6×3 and 6³?

$6 \times 3 = 18$ (multiplication), but $6^3 = 6\cdot6\cdot6 = 216$ (repeated multiplication). The exponent tells you how many times to multiply the base by itself.

Can I just multiply 6×6×6 to check my answer?

Absolutely! $6\times6\times6 = 216$ , and $6^3 = 216$ too. If both expressions give the same result, your exponent is correct.

What if there were more 6's in the multiplication?

Just keep counting! If you had $6\cdot6\cdot6\cdot6\cdot6$ , that would be $6^5$ because the base 6 appears 5 times.

Why can't I just look at the answer 216 to find n?

While you could work backwards, it's much easier to count the factors directly. Plus, this method works even when the numbers get really big and hard to calculate!

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