Comparing Mathematical Expressions: Finding Greater Value When b > 1

Q: What happens with negative exponents like b^(-2)?

Negative exponents still follow the same rules! $ b^3 \times b^{-2} = b^{3+(-2)} = b^1 = b $. Think of negative exponents as subtracting from the total power.

Q: How do I know which expression is greater when b > 1?

When the base $ b > 1 $, larger exponents mean larger values. So $ b^{11} > b^7 > b^6 > b^3 $. If b were between 0 and 1, it would be the opposite!

Q: Do I need to calculate actual numbers?

No! Since all expressions have the same base $ b $, you only need to compare the final exponents. The expression with the highest exponent wins when $ b > 1 $.

Q: What if I forgot the exponent rules?

$ a^m \times a^n = a^{m+n} $ (add exponents)$ (a^m)^n = a^{m \times n} $ (multiply exponents)$ a^0 = 1 $ (any base to power 0 equals 1)Practice these rules until they become automatic!

Exponent Rules with Multiple Operations

Which expression has the greater value given that $b>1$ ?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine the largest value

00:03 When multiplying powers with equal bases

00:06 The power of the result equals the sum of the powers

00:09 We'll apply this formula to our exercise and add the powers together

00:12 We'll use this formula in order to calculate all the powers

00:27 We'll choose the largest power, and that's the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which expression has the greater value given that $b>1$ ?

Step-by-step solution

To solve this problem, let's simplify and compare the given expressions one by one.

Simplification of each expression:
$b^3 \times b^5 \times b^{-2} = b^{3+5-2} = b^6$
$b^7 \times b^4 = b^{7+4} = b^{11}$
$(b)^3 \times b^4 = b^{3+4} = b^7$
$b^{-3} \times b^6 = b^{-3+6} = b^3$

Next, we compare the simplified exponents:
- The first expression simplifies to $b^6$ .
- The second expression simplifies to $b^{11}$ .
- The third expression simplifies to $b^7$ .
- The fourth expression simplifies to $b^3$ .

Among these, $b^{11}$ is the greatest because exponent 11 is the highest. Since $b > 1$ , greater exponents correspond to greater values.

Therefore, the expression with the greatest value is $b^7 \times b^4$ , which corresponds to choice 2.

Final Answer

$b^7\times b^4$

Key Points to Remember

Essential concepts to master this topic

Rule: When multiplying powers with same base, add the exponents
Technique: Simplify $b^7 \times b^4 = b^{7+4} = b^{11}$
Check: Compare final exponents since b > 1 means higher exponent equals greater value ✓

Common Mistakes

Avoid these frequent errors

Not simplifying expressions before comparing
Don't compare $b^3 \times b^5 \times b^{-2}$ with $b^7 \times b^4$ without simplifying = wrong conclusion! You might think the first is bigger because it has more terms. Always simplify each expression completely first, then compare the final exponents.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do I add exponents when multiplying?

When you multiply powers with the same base, you're really combining repeated multiplication. For example, $b^3 \times b^2 = (b \times b \times b) \times (b \times b) = b^5$ . The exponent addition rule is just a shortcut!

What happens with negative exponents like b^(-2)?

Negative exponents still follow the same rules! $b^3 \times b^{-2} = b^{3+(-2)} = b^1 = b$ . Think of negative exponents as subtracting from the total power.

How do I know which expression is greater when b > 1?

When the base $b > 1$ , larger exponents mean larger values. So $b^{11} > b^7 > b^6 > b^3$ . If b were between 0 and 1, it would be the opposite!

Do I need to calculate actual numbers?

No! Since all expressions have the same base $b$ , you only need to compare the final exponents. The expression with the highest exponent wins when $b > 1$ .

What if I forgot the exponent rules?

$a^m \times a^n = a^{m+n}$ (add exponents)
$(a^m)^n = a^{m \times n}$ (multiply exponents)
$a^0 = 1$ (any base to power 0 equals 1)

Practice these rules until they become automatic!

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