Simplify the Expression: 6^(4x) ÷ 6^(x+1)

Q: What if the exponent in the denominator is larger than in the numerator?

No problem! You'll get a negative exponent. For example, $ 6^2 ÷ 6^5 = 6^{2-5} = 6^{-3} $. Remember that negative exponents mean the reciprocal: $ 6^{-3} = \frac{1}{6^3} $.

Q: How do I handle parentheses in the exponent subtraction?

Always use the distributive property carefully! When you see 4x - (x + 1), the negative sign distributes to everything inside: $ 4x - x - 1 = 3x - 1 $. Don't forget that minus sign affects all terms!

Q: Can I use this rule with different bases like 6^x ÷ 3^x?

No! The quotient rule only works when the bases are identical. For $ 6^x ÷ 3^x $, you'd need to rewrite it as $ (6÷3)^x = 2^x $ using different rules.

Q: What's the difference between 6^(3x-1) and 6^3x-1?

The parentheses are crucial! $ 6^{3x-1} $ means the entire expression (3x-1) is the exponent. But $ 6^{3x} - 1 $ means you raise 6 to the 3x power, then subtract 1. Very different results!

Quotient Rule with Variable Exponents

Insert the corresponding expression:

$\frac{6^{4x}}{6^{x+1}}=$

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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 number (A) to the power of (N) divided by the same base (A) to the power of (M)

00:07 equals the number (A) to the power of the difference of exponents (M-N)

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

00:12 Let's properly expand the parentheses

00:15 Negative times positive always equals negative

00:19 Let's group like terms

00:22 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:

$\frac{6^{4x}}{6^{x+1}}=$

Step-by-step solution

To solve the given expression $\frac{6^{4x}}{6^{x+1}}$ , we must apply the Power of a Quotient Rule for Exponents. This rule states that $\frac{a^m}{a^n} = a^{m-n}$ .

Using this rule, the given expression can be rewritten as follows:

The numerator is $6^{4x}$ .
The denominator is $6^{x+1}$ .

Apply the Power of a Quotient Rule:

$\frac{6^{4x}}{6^{x+1}} = 6^{4x - (x + 1)}$

We need to simplify the exponent by performing the subtraction $4x - (x + 1)$ :

Step 1: Distribute the subtraction sign to the terms inside the parenthesis:

$4x - x - 1$

Step 2: Combine like terms:

$3x - 1$

The expression simplifies to:

$6^{3x-1}$

Therefore, the solution to the question is: $6^{3x-1}$ .

Final Answer

$6^{3x-1}$

Key Points to Remember

Essential concepts to master this topic

Rule: For division of same bases, subtract exponents: a^m ÷ a^n = a^(m-n)
Technique: Carefully distribute negative sign: 4x - (x+1) = 4x - x - 1
Check: Verify by expanding: 6^(3x-1) = 6^4x ÷ 6^(x+1) when substituted ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the negative sign in subtraction
Don't write 4x - (x + 1) as 4x - x + 1 = 3x + 1! This gives the wrong exponent because you forgot the negative distributes to both terms. Always distribute the negative: 4x - (x + 1) = 4x - x - 1 = 3x - 1.

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 powers with the same base?

When you divide powers with the same base, you're essentially canceling out common factors. For example, $6^4 ÷ 6^2 = \frac{6×6×6×6}{6×6}$ , and two 6's cancel out, leaving $6^2$ . That's why we subtract: 4 - 2 = 2!

What if the exponent in the denominator is larger than in the numerator?

No problem! You'll get a negative exponent. For example, $6^2 ÷ 6^5 = 6^{2-5} = 6^{-3}$ . Remember that negative exponents mean the reciprocal: $6^{-3} = \frac{1}{6^3}$ .

How do I handle parentheses in the exponent subtraction?

Always use the distributive property carefully! When you see 4x - (x + 1), the negative sign distributes to everything inside: $4x - x - 1 = 3x - 1$ . Don't forget that minus sign affects all terms!

Can I use this rule with different bases like 6^x ÷ 3^x?

No! The quotient rule only works when the bases are identical. For $6^x ÷ 3^x$ , you'd need to rewrite it as $(6÷3)^x = 2^x$ using different rules.

What's the difference between 6^(3x-1) and 6^3x-1?

The parentheses are crucial! $6^{3x-1}$ means the entire expression (3x-1) is the exponent. But $6^{3x} - 1$ means you raise 6 to the 3x power, then subtract 1. Very different results!

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