Simplify the Expression: 7^10 Divided by 7^13 Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it as canceling out common factors! When you divide $ 7^{10} \div 7^{13} $, you're canceling 10 sevens from both top and bottom, leaving $ \frac{1}{7^3} $.

Q: What does a negative exponent really mean?

A negative exponent means "flip and make positive"! So $ 7^{-3} = \frac{1}{7^3} $. It's like moving the base from numerator to denominator.

Q: Can I just cancel out the 7s directly?

Not directly! You can't cross out bases like regular numbers. You must use the quotient rule: subtract exponents when bases are the same.

Q: Why isn't the answer just 7^(10/13)?

That would be if you were taking a root, not dividing! Division of powers uses subtraction: $ \frac{a^m}{a^n} = a^{m-n} $, not $ a^{m/n} $.

Q: How do I remember when to add vs subtract exponents?

Multiplication = Add exponents, Division = Subtract exponents. Think: "Same operation, same base → combine exponents!"

Q: Is 1/7³ the final answer or should I calculate 343?

Leave it as $ \frac{1}{7^3} $! This exact form is usually preferred over the decimal approximation unless specifically asked to calculate.

Exponent Rules with Negative Powers

Insert the corresponding expression:

$\frac{7^{10}}{7^{13}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Alright, let's dive into it.

00:12 We're using a formula for dividing exponents.

00:16 When a number, A, to the power of N is divided by A to the power of M,

00:22 it becomes A to the power of M minus N.

00:25 We'll apply this rule in our exercise.

00:29 Now, let's use the formula for negative exponents.

00:33 A to the power of negative N becomes one over A to the power of N.

00:39 Remember, it's the reciprocal with the opposite power.

00:43 We'll use this in our exercise too.

00:46 Substitute the reciprocal and the opposite power.

00:50 And there you go, that's how we solve it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

$\frac{7^{10}}{7^{13}}=$

Step-by-step solution

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

Step 1: Apply the quotient rule for exponents
Step 2: Simplify the resulting expression
Step 3: Compare the simplified form with the given choices

Now, let's work through each step:
Step 1: Given the expression $\frac{7^{10}}{7^{13}}$ , use the quotient rule for exponents, which states $\frac{a^m}{a^n} = a^{m-n}$ .
Step 2: Apply this rule to get $7^{10-13} = 7^{-3}$ .
Step 3: Rewrite $7^{-3}$ using the rule for negative exponents, which is $a^{-n} = \frac{1}{a^n}$ . Therefore, $7^{-3} = \frac{1}{7^3}$ .

Comparing with the provided answer choices, the correct choice is:

Choice 2: $\frac{1}{7^3}$

Therefore, the solution to the problem is $\frac{1}{7^3}$ , confirming the correctness of the derived expression and matching the provided answer.

Final Answer

$\frac{1}{7^3}$

Key Points to Remember

Essential concepts to master this topic

Quotient Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{7^{10}}{7^{13}} = 7^{10-13} = 7^{-3}$
Check: Convert negative exponent: $7^{-3} = \frac{1}{7^3}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting when dividing
Don't add 10 + 13 = 23 when dividing = $7^{23}$ ! This confuses multiplication rules with division rules. Always subtract the bottom exponent from the top exponent when dividing same bases.

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 as canceling out common factors! When you divide $7^{10} \div 7^{13}$ , you're canceling 10 sevens from both top and bottom, leaving $\frac{1}{7^3}$ .

What does a negative exponent really mean?

A negative exponent means "flip and make positive"! So $7^{-3} = \frac{1}{7^3}$ . It's like moving the base from numerator to denominator.

Can I just cancel out the 7s directly?

Not directly! You can't cross out bases like regular numbers. You must use the quotient rule: subtract exponents when bases are the same.

Why isn't the answer just 7^(10/13)?

That would be if you were taking a root, not dividing! Division of powers uses subtraction: $\frac{a^m}{a^n} = a^{m-n}$ , not $a^{m/n}$ .

How do I remember when to add vs subtract exponents?

Multiplication = Add exponents, Division = Subtract exponents. Think: "Same operation, same base → combine exponents!"

Is 1/7³ the final answer or should I calculate 343?

Leave it as $\frac{1}{7^3}$ ! This exact form is usually preferred over the decimal approximation unless specifically asked to calculate.

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