Solve: Simplifying 14a^-3 ÷ 7a^-3 with Like Terms

Q: Why do I subtract the exponents when dividing?

The quotient rule for exponents states $ \frac{a^m}{a^n} = a^{m-n} $. Think of it as canceling: $ a^{-3} ÷ a^{-3} $ means the same power cancels completely, leaving $ a^0 = 1 $.

Q: What happens when I have negative exponents like -3?

Negative exponents work the same way! When subtracting: $ -3 - (-3) = -3 + 3 = 0 $. Remember that subtracting a negative is the same as adding.

Q: Why does anything to the power of 0 equal 1?

This is a fundamental rule: $ a^0 = 1 $ for any non-zero number a. Think of it as complete cancellation - when identical powers divide, they cancel out completely, leaving just 1.

Q: Do I handle the coefficients differently from the variables?

Yes! Always separate them: divide the numbers normally ($ \frac{14}{7} = 2 $) and apply exponent rules to variables separately ($ \frac{a^{-3}}{a^{-3}} = a^0 $).

Q: What if the exponents were different, like -2 and -3?

Use the same rule: $ \frac{a^{-2}}{a^{-3}} = a^{-2-(-3)} = a^{-2+3} = a^1 = a $. The key is careful subtraction of the exponents!

Exponent Division with Identical Bases

Solve the following exercise:

$\frac{14a^{-3}}{7a^{-3}}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When dividing powers with equal bases

00:07 The power of the result equals the difference between the powers

00:11 We'll apply this formula to our exercise and subtract the powers

00:16 Let's divide the numerical fraction and the unknown fraction

00:25 Let's calculate 14 divided by 7

00:32 Any number raised to the power of 0 is always equal to 1

00:37 As long as the number is not 0

00:40 We'll apply this formula to our exercise and substitute in 1

00:45 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\frac{14a^{-3}}{7a^{-3}}=$

Step-by-step solution

Due to the fact that the numerator and the denominator of the fraction have terms with identical bases, we will begin by applying the law of exponents for the division of terms with identical bases:

$\frac{b^m}{b^n}=b^{m-n}$ We apply it to the problem:

$\frac{14a^{-3}}{7a^{-3}}=2a^{-3-(-3)}=2a^{-3+3}=2a^0$ In the first step we simplify the numerical part of the fraction. This is a simple and intuitive step as it makes it easier to work with the said fraction.:

$\frac{14a^{-3}}{7a^{-3}}=\frac{14}{7}\cdot\frac{a^{-3}}{a^{-3}}=2a^{-3-(-3)}=\ldots$ We then return to the problem and remember that any number raised to the 0th power is 1, that is:

$b^0=1$ Thus, in the problem we obtain the following:

$2a^0=2\cdot1=2$ Therefore, the correct answer is option B.

Final Answer

$2$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing powers with same base, subtract exponents: $\frac{b^m}{b^n} = b^{m-n}$
Technique: Separate coefficients first: $\frac{14}{7} \cdot \frac{a^{-3}}{a^{-3}} = 2 \cdot a^{-3-(-3)}$
Check: Verify $a^0 = 1$ so $2a^0 = 2 \cdot 1 = 2$ ✓

Common Mistakes

Avoid these frequent errors

Multiplying exponents instead of subtracting
Don't multiply -3 × -3 = 9 to get $2a^9$ ! Division means subtract exponents, not multiply them. Always use $a^{-3} ÷ a^{-3} = a^{-3-(-3)} = a^0$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I subtract the exponents when dividing?

The quotient rule for exponents states $\frac{a^m}{a^n} = a^{m-n}$ . Think of it as canceling: $a^{-3} ÷ a^{-3}$ means the same power cancels completely, leaving $a^0 = 1$ .

What happens when I have negative exponents like -3?

Negative exponents work the same way! When subtracting: $-3 - (-3) = -3 + 3 = 0$ . Remember that subtracting a negative is the same as adding.

Why does anything to the power of 0 equal 1?

This is a fundamental rule: $a^0 = 1$ for any non-zero number a. Think of it as complete cancellation - when identical powers divide, they cancel out completely, leaving just 1.

Do I handle the coefficients differently from the variables?

Yes! Always separate them: divide the numbers normally ( $\frac{14}{7} = 2$ ) and apply exponent rules to variables separately ( $\frac{a^{-3}}{a^{-3}} = a^0$ ).

What if the exponents were different, like -2 and -3?

Use the same rule: $\frac{a^{-2}}{a^{-3}} = a^{-2-(-3)} = a^{-2+3} = a^1 = a$ . The key is careful subtraction of the exponents!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime