Simplify the Expression: b^10 × b^-5 ÷ (b^11/b^6)

Q: Why do I multiply by the reciprocal instead of dividing directly?

Dividing by a fraction is the same as multiplying by its reciprocal. This makes the calculation much easier! Think of it like: $ ÷ \frac{2}{3} = × \frac{3}{2} $.

Q: How do I remember which exponent rule to use?

Multiplication: add exponents ($ b^3 × b^2 = b^5 $)Division: subtract exponents ($ b^5 ÷ b^2 = b^3 $)Just remember: same operation as with the exponents!

Q: Can I simplify the expression in a different order?

Yes! You could simplify $ \frac{b^{11}}{b^6} $ first to get $ b^5 $, then work with $ b^{10} × b^{-5} ÷ b^5 $. The final answer will be the same.

Q: What if the base was a number instead of a variable?

The same rules apply! For example: $ 2^{10} × 2^{-5} ÷ \frac{2^{11}}{2^6} = 2^0 = 1 $. Exponent laws work with any base.

Exponent Laws with Division Operations

Simplify the following:

$b^{10}\times b^{-5}:\frac{b^{11}}{b^6}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 When multiplying powers with equal bases

00:07 The power of the result equals the sum of the powers

00:10 We'll apply this formula to our exercise, and add up the powers

00:25 When dividing powers with equal bases

00:28 The power of the result equals the difference of the powers

00:31 We'll apply this formula to our exercise, and then subtract the powers

00:51 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Simplify the following:

$b^{10}\times b^{-5}:\frac{b^{11}}{b^6}=$

Step-by-step solution

To simplify the expression $b^{10} \times b^{-5} \div \frac{b^{11}}{b^6}$ , we will follow a systematic approach.

First, simplify the numerator:

We have $b^{10} \times b^{-5}$ . Using the rule for multiplying powers with the same base, we get:

$b^{10 + (-5)} = b^{5}$

Next, simplify the expression in the denominator:

The denominator is $\frac{b^{11}}{b^6}$ . Using the rule for dividing powers with the same base, we have:

$b^{11 - 6} = b^{5}$

Now, divide the simplified numerator by the simplified denominator:

$\frac{b^5}{b^5} = b^{5-5} = b^0$

We know that any non-zero number raised to the power of 0 is 1, therefore:

$b^0 = 1$

Therefore, the simplified expression is $1$ .

Final Answer

$1$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying same bases, add exponents together
Division Method: Convert $÷ \frac{b^{11}}{b^6}$ to $× \frac{b^6}{b^{11}}$
Check: Final result $b^0 = 1$ because any base to power 0 equals 1 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to flip the fraction when dividing
Don't divide by $\frac{b^{11}}{b^6}$ directly = wrong exponent calculations! Division by a fraction means multiplying by its reciprocal. Always flip the fraction: $÷ \frac{b^{11}}{b^6} = × \frac{b^6}{b^{11}}$ .

Practice Quiz

Test your knowledge with interactive questions

Simplify the following equation:

$$ $4^5\times4^5=$

FAQ

Everything you need to know about this question

Why do I multiply by the reciprocal instead of dividing directly?

Dividing by a fraction is the same as multiplying by its reciprocal. This makes the calculation much easier! Think of it like: $÷ \frac{2}{3} = × \frac{3}{2}$ .

How do I remember which exponent rule to use?

Multiplication: add exponents ( $b^3 × b^2 = b^5$ )
Division: subtract exponents ( $b^5 ÷ b^2 = b^3$ )
Just remember: same operation as with the exponents!

What does b⁰ really mean?

Any non-zero number raised to the power of 0 equals 1. Think of it as $\frac{b^5}{b^5} = 1$ - you're dividing something by itself!

Can I simplify the expression in a different order?

Yes! You could simplify $\frac{b^{11}}{b^6}$ first to get $b^5$ , then work with $b^{10} × b^{-5} ÷ b^5$ . The final answer will be the same.

What if the base was a number instead of a variable?

The same rules apply! For example: $2^{10} × 2^{-5} ÷ \frac{2^{11}}{2^6} = 2^0 = 1$ . Exponent laws work with any base.

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Simplify the Expression: b^10 × b^-5 ÷ (b^11/b^6)

Exponent Laws with Division Operations

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I multiply by the reciprocal instead of dividing directly?

How do I remember which exponent rule to use?

What does b⁰ really mean?

Can I simplify the expression in a different order?

What if the base was a number instead of a variable?

🌟 Unlock Your Math Potential

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