Simplify the Expression: 4^5 Divided by 4^x

Q: Why do we subtract exponents when dividing?

Think of it this way: $ \frac{4^5}{4^x} $ means "how many times does $ 4^x $ go into $ 4^5 $?" The answer is $ 4^{5-x} $ because you're removing x factors of 4 from the 5 factors.

Q: Can I use this rule with different bases?

No! This rule only works when the bases are exactly the same. You cannot use it for something like $ \frac{3^5}{4^x} $ because 3 and 4 are different bases.

Q: What happens if x equals 5?

Great question! If x = 5, then $ \frac{4^5}{4^5} = 4^{5-5} = 4^0 = 1 $. Any number divided by itself equals 1!

Q: How is this different from multiplying exponents?

When multiplying same bases, you add exponents: $ 4^a \cdot 4^b = 4^{a+b} $. When dividing same bases, you subtract exponents: $ \frac{4^a}{4^b} = 4^{a-b} $.

Exponent Division with Variable Powers

Insert the corresponding expression:

$\frac{4^5}{4^x}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simply

00:02 Let's use the formula for dividing exponents

00:04 Any number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:06 equals number (A) to the power of the difference of exponents (M-N)

00:08 Let's use this formula in our exercise

00:10 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:

$\frac{4^5}{4^x}=$

Step-by-step solution

We need to simplify the expression $\frac{4^5}{4^x}$ .

According to the rules of exponents, specifically the power of a quotient rule, when you divide like bases you subtract the exponents. The rule is written as:

$\frac{a^m}{a^n} = a^{m-n}$

This means we take the exponent in the numerator and subtract the exponent in the denominator. Let's apply this rule to our expression:

$\frac{4^5}{4^x} = 4^{5-x}$

Hence, the simplified form of the expression is $4^{5-x}$ .

The solution to the question is: $4^{5-x}$

Final Answer

$4^{5-x}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract the exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: For $\frac{4^5}{4^x}$ , subtract: 5 - x = $4^{5-x}$
Check: Verify by expanding: $4^5 ÷ 4^x = 4^5 \cdot 4^{-x} = 4^{5-x}$ ✓

Common Mistakes

Avoid these frequent errors

Subtracting exponents in wrong order
Don't write $4^{x-5}$ instead of $4^{5-x}$ = completely wrong answer! The order matters because subtraction isn't commutative. Always subtract the denominator exponent from the numerator exponent: top minus bottom.

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do we subtract exponents when dividing?

Think of it this way: $\frac{4^5}{4^x}$ means "how many times does $4^x$ go into $4^5$ ?" The answer is $4^{5-x}$ because you're removing x factors of 4 from the 5 factors.

What if x is bigger than 5?

That's totally fine! If x > 5, then 5-x will be negative, giving you $4^{\text{negative}}$ . For example, if x = 7, you get $4^{5-7} = 4^{-2} = \frac{1}{4^2}$ .

Can I use this rule with different bases?

No! This rule only works when the bases are exactly the same. You cannot use it for something like $\frac{3^5}{4^x}$ because 3 and 4 are different bases.

What happens if x equals 5?

Great question! If x = 5, then $\frac{4^5}{4^5} = 4^{5-5} = 4^0 = 1$ . Any number divided by itself equals 1!

How is this different from multiplying exponents?

When multiplying same bases, you add exponents: $4^a \cdot 4^b = 4^{a+b}$ . When dividing same bases, you subtract exponents: $\frac{4^a}{4^b} = 4^{a-b}$ .

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