Simplify 6^7 ÷ 6^4: Applying Laws of Exponents

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{6^7}{6^4} $ means how many times does $ 6^4 $ go into $ 6^7 $? Since $ 6^7 = 6^4 \times 6^3 $, the answer is $ 6^3 $.

Q: What if the bottom exponent is bigger than the top?

You still subtract! For example, $ \frac{6^4}{6^7} = 6^{4-7} = 6^{-3} $. The negative exponent means one over that positive power: $ \frac{1}{6^3} $.

Q: Do the bases have to be exactly the same?

Yes! The quotient rule only works when the bases are identical. You cannot use it for $ \frac{6^7}{5^4} $ because 6 ≠ 5.

Q: Can I use this rule with variables too?

Absolutely! $ \frac{x^7}{x^4} = x^{7-4} = x^3 $ works the same way. The base can be any number or variable, as long as it's the same on top and bottom.

Q: What if one of the exponents is 1?

The rule still applies! For example, $ \frac{6^7}{6^1} = 6^{7-1} = 6^6 $. Remember that $ 6^1 = 6 $, so this is the same as $ \frac{6^7}{6} $.

Quotient Rule with Same Base Division

Insert the corresponding expression:

$\frac{6^7}{6^4}=$

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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 dividing powers

00:04 Any division of powers with the same base (A) and different exponents

00:07 equals the same base (A) raised to the difference of the exponents (M-N)

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

00:13 And we'll compare the numbers to the variables in the formula

00:27 We'll keep the base and subtract between the exponents

00:42 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

Insert the corresponding expression:

$\frac{6^7}{6^4}=$

Step-by-step solution

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

Identify the given information and relevant exponent rules.
Apply the quotient property of exponents.
Simplify the expression.

Now, let's work through each step:
Step 1: The problem gives us the expression $\frac{6^7}{6^4}$ . The base is 6, and the exponents are 7 and 4, respectively.
Step 2: According to the rule of exponents, when dividing powers with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$ In this case, $a = 6$ , $m = 7$ , and $n = 4$ .
Step 3: Applying this rule gives us: $\frac{6^7}{6^4} = 6^{7 - 4} = 6^3$

Therefore, the solution to the problem is $6^3$ .

Final Answer

$6^3$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing same bases, subtract the exponents
Technique: $\frac{6^7}{6^4} = 6^{7-4} = 6^3$
Check: Verify $6^3 = 216$ and $\frac{6^7}{6^4} = \frac{279936}{1296} = 216$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting
Don't add the exponents like 7 + 4 = 11 to get $6^{11}$ ! This gives the wrong rule for multiplication, not division. Always remember: division means subtract exponents, multiplication means add exponents.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{6^7}{6^4}$ means how many times does $6^4$ go into $6^7$ ? Since $6^7 = 6^4 \times 6^3$ , the answer is $6^3$ .

What if the bottom exponent is bigger than the top?

You still subtract! For example, $\frac{6^4}{6^7} = 6^{4-7} = 6^{-3}$ . The negative exponent means one over that positive power: $\frac{1}{6^3}$ .

Do the bases have to be exactly the same?

Yes! The quotient rule only works when the bases are identical. You cannot use it for $\frac{6^7}{5^4}$ because 6 ≠ 5.

Can I use this rule with variables too?

Absolutely! $\frac{x^7}{x^4} = x^{7-4} = x^3$ works the same way. The base can be any number or variable, as long as it's the same on top and bottom.

What if one of the exponents is 1?

The rule still applies! For example, $\frac{6^7}{6^1} = 6^{7-1} = 6^6$ . Remember that $6^1 = 6$ , so this is the same as $\frac{6^7}{6}$ .

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