Solve Exponential Fraction: b^y/b^x - b^z/b^3 Simplification

Q: Why do I subtract the exponents when dividing?

This comes from the quotient rule for exponents: $ \frac{a^m}{a^n} = a^{m-n} $. Think of it as canceling out common factors - if you have $ \frac{b \times b \times b}{b \times b} $, you cancel two b's and get one b left!

Q: Do I need the same base for this rule to work?

Yes, absolutely! The quotient rule only works when the bases are identical. You cannot simplify $ \frac{a^3}{b^2} $ using this rule because a and b are different bases.

Q: What happens to the minus sign in the middle?

The minus sign stays exactly the same! You're simplifying each fraction separately, but the subtraction operation between them doesn't change. So $ \frac{b^y}{b^x} - \frac{b^z}{b^3} $ becomes $ b^{y-x} - b^{z-3} $.

Q: Can I combine the terms after simplifying?

No, you cannot! Terms like $ b^{y-x} $ and $ b^{z-3} $ have different exponents, so they are not like terms. You can only add or subtract terms with identical bases and identical exponents.

Q: What if one of the exponents is just a number?

It works the same way! In this problem, $ b^3 $ has exponent 3, so $ \frac{b^z}{b^3} = b^{z-3} $. The rule applies whether exponents are variables or numbers.

Exponent Division with Subtraction Operations

Solve the following:

$\frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 Let's simplify this problem!

00:14 When dividing powers with the same base, remember,

00:19 The power of the result is the power at the top minus the power at the bottom.

00:24 Let's use this rule now and subtract those powers.

00:27 And there you go, we've solved it!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following:

$\frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=$

Step-by-step solution

Here we have division between two terms with identical bases, therefore we will use the power property to divide terms with identical bases:

$\frac{c^m}{c^n}=c^{m-n}$ Note that using this property is only possible when the division is carried out between terms with identical bases.

Let's go back to the problem and apply the power property to each term of the exercise separately:

$\frac{b^{\frac{y}{}}}{b^x}-\frac{b^z}{b^3}=b^{y-x}-b^{z-3}$ Therefore, the correct answer is option A.

Final Answer

$b^{y-x}-b^{z-3}$

Key Points to Remember

Essential concepts to master this topic

Rule: When dividing same bases, subtract exponents: $\frac{a^m}{a^n} = a^{m-n}$
Technique: Apply division rule separately: $\frac{b^y}{b^x} = b^{y-x}$ and $\frac{b^z}{b^3} = b^{z-3}$
Check: Final expression keeps subtraction sign: $b^{y-x} - b^{z-3}$ ✓

Common Mistakes

Avoid these frequent errors

Changing the subtraction sign to addition
Don't write $b^{y-x} + b^{z-3}$ when the original has subtraction = wrong sign in final answer! Students often forget that the subtraction between the two fractions stays the same after simplifying each fraction. Always keep the original operation sign between terms.

Practice Quiz

Test your knowledge with interactive questions

Insert the corresponding expression:

$\frac{9^{11}}{9^4}=$

FAQ

Everything you need to know about this question

Why do I subtract the exponents when dividing?

This comes from the quotient rule for exponents: $\frac{a^m}{a^n} = a^{m-n}$ . Think of it as canceling out common factors - if you have $\frac{b \times b \times b}{b \times b}$ , you cancel two b's and get one b left!

Do I need the same base for this rule to work?

Yes, absolutely! The quotient rule only works when the bases are identical. You cannot simplify $\frac{a^3}{b^2}$ using this rule because a and b are different bases.

What happens to the minus sign in the middle?

The minus sign stays exactly the same! You're simplifying each fraction separately, but the subtraction operation between them doesn't change. So $\frac{b^y}{b^x} - \frac{b^z}{b^3}$ becomes $b^{y-x} - b^{z-3}$ .

Can I combine the terms after simplifying?

No, you cannot! Terms like $b^{y-x}$ and $b^{z-3}$ have different exponents, so they are not like terms. You can only add or subtract terms with identical bases and identical exponents.

What if one of the exponents is just a number?

It works the same way! In this problem, $b^3$ has exponent 3, so $\frac{b^z}{b^3} = b^{z-3}$ . The rule applies whether exponents are variables or numbers.

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