Simplify 7^5b/7^2b: Exponent Division with Like Bases

Exponent Division with Algebraic Expressions

Insert the corresponding expression:

75b72b= \frac{7^{5b}}{7^{2b}}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:08 Let's start simply.
00:11 We're using a formula for dividing powers.
00:14 If you have a number, A, to the power of N, divided by A to the power of M,
00:20 it equals A to the power of M minus N.
00:24 We'll apply this formula to our exercise.
00:29 And that's how we solve the question!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Insert the corresponding expression:

75b72b= \frac{7^{5b}}{7^{2b}}=

2

Step-by-step solution

To solve the expression 75b72b \frac{7^{5b}}{7^{2b}} , we will use the Power of a Quotient Rule for Exponents, which states that aman=amn \frac{a^m}{a^n} = a^{m-n} when a a is a nonzero number. This rule allows us to simplify expressions where the bases are the same.

1. Identify the base and the exponents in the expression. Here, the base is 7, and the exponents are 5b 5b and 2b 2b .

2. Apply the Power of a Quotient Rule:
75b72b=75b2b \frac{7^{5b}}{7^{2b}} = 7^{5b - 2b}

3. Simplify the expression in the exponent:
Calculate 5b2b=3b 5b - 2b = 3b .

4. Therefore, the expression simplifies to 73b 7^{3b} .

However, according to the given correct answer, we are asked to provide the intermediate expression as well – that is, before calculating the difference:
So, the solution as an intermediate step is:
75b2b 7^{5b - 2b}

The explicit step-by-step answer provided in the question's solution matches our intermediate form.

The solution to the question is:

75b2b 7^{5b - 2b}

3

Final Answer

75b2b 7^{5b-2b}

Key Points to Remember

Essential concepts to master this topic
  • Division Rule: For same bases, subtract exponents: aman=amn \frac{a^m}{a^n} = a^{m-n}
  • Technique: Apply subtraction: 75b72b=75b2b \frac{7^{5b}}{7^{2b}} = 7^{5b-2b}
  • Check: Verify bases match before subtracting exponents ✓

Common Mistakes

Avoid these frequent errors
  • Adding exponents instead of subtracting during division
    Don't add exponents like 75b+2b=77b 7^{5b+2b} = 7^{7b} ! This applies multiplication rules to division and gives completely wrong results. Always subtract the bottom exponent from the top exponent when dividing like bases.

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?

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Think of it as canceling out common factors! 75b72b \frac{7^{5b}}{7^{2b}} means you have 5b sevens in the numerator and 2b sevens in the denominator. After canceling 2b sevens, you're left with (5b-2b) = 3b sevens.

Can I simplify 75b2b 7^{5b-2b} further?

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Yes! You can combine like terms: 5b2b=3b 5b - 2b = 3b , so the final answer is 73b 7^{3b} . However, some problems ask for the intermediate step 75b2b 7^{5b-2b} to show your work.

What if the bases were different numbers?

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You cannot use this rule! The division rule aman=amn \frac{a^m}{a^n} = a^{m-n} only works when the bases are exactly the same. Different bases require different methods.

Do I subtract exponents when multiplying too?

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No! When multiplying same bases, you add exponents: aman=am+n a^m \cdot a^n = a^{m+n} . Only subtract when dividing. Remember: multiply = add, divide = subtract.

What if one of the exponents is bigger?

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It doesn't matter! Always subtract bottom from top: 72b75b=72b5b=73b \frac{7^{2b}}{7^{5b}} = 7^{2b-5b} = 7^{-3b} . Negative exponents are perfectly valid and mean 173b \frac{1}{7^{3b}} .

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