Simplify the Expression: Finding a⁵÷a⁴ Using Exponent Rules

Q: Why do we subtract exponents when dividing?

Think of it as canceling! $ \frac{a^5}{a^4} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a \cdot a} $. Four a's cancel out, leaving just one a.

Q: What if the exponents are equal?

When exponents are equal, like $ a^3 ÷ a^3 $, you get $ a^{3-3} = a^0 = 1 $. Any non-zero number to the zero power equals 1!

Q: Does this work with negative exponents too?

Yes! The rule works for all exponents. For example: $ a^2 ÷ a^5 = a^{2-5} = a^{-3} = \frac{1}{a^3} $. Just subtract normally.

Q: What does the colon symbol mean in math?

The colon (:) is another way to write division, just like ÷. So $ a^5:a^4 $ means the same as $ a^5 ÷ a^4 $ or $ \frac{a^5}{a^4} $.

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

Multiplication = Add exponents: $ a^2 \cdot a^3 = a^5 $Division = Subtract exponents: $ a^5 ÷ a^3 = a^2 $Think: More operations (multiply) = add, Less stuff (divide) = subtract!

Exponent Division Rules with Same Base

Choose the expression that is equal to the following:

$a^5:a^4$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 Let's write the division as a fraction

00:09 When dividing powers with equal bases

00:12 The power of the result equals the difference between the exponents

00:17 We'll apply this formula to our exercise, and subtract the exponents

00:27 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that is equal to the following:

$a^5:a^4$

Step-by-step solution

First, for good order, let's write the expression as a fraction:

$\frac{a^5}{a^4}$ Then we'll recall the law of exponents for division between terms with equal bases:

$\frac{c^m}{c^n}=c^{m-n}$ and we'll apply this law to our problem:

$\frac{a^5}{a^4}=a^{5-4}=a^1=a$ where in the second step we calculated the result of the subtraction in the exponent and then used the fact that any number raised to the power of 1 equals the number itself, meaning that:

$X^1=X$ We got that: $\frac{a^5}{a^4}=a$ therefore the correct answer is a.

Final Answer

$a$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base, subtract exponents
Technique: $a^5 ÷ a^4 = a^{5-4} = a^1 = a$
Check: Verify $a^1 = a$ since any number to first power equals itself ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of subtracting during division
Don't add the exponents 5 + 4 = 9 to get $a^9$ ! This gives multiplication, not division. Always subtract exponents when dividing same bases: $a^5 ÷ a^4 = a^{5-4} = a$ .

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! $\frac{a^5}{a^4} = \frac{a \cdot a \cdot a \cdot a \cdot a}{a \cdot a \cdot a \cdot a}$ . Four a's cancel out, leaving just one a.

What if the exponents are equal?

When exponents are equal, like $a^3 ÷ a^3$ , you get $a^{3-3} = a^0 = 1$ . Any non-zero number to the zero power equals 1!

Does this work with negative exponents too?

Yes! The rule works for all exponents. For example: $a^2 ÷ a^5 = a^{2-5} = a^{-3} = \frac{1}{a^3}$ . Just subtract normally.

What does the colon symbol mean in math?

The colon (:) is another way to write division, just like ÷. So $a^5:a^4$ means the same as $a^5 ÷ a^4$ or $\frac{a^5}{a^4}$ .

How do I remember when to add vs subtract exponents?

Multiplication = Add exponents: $a^2 \cdot a^3 = a^5$
Division = Subtract exponents: $a^5 ÷ a^3 = a^2$
Think: More operations (multiply) = add, Less stuff (divide) = subtract!

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