Mathematical Comparison: Determining the Greater Value Between Two Numbers

Q: Why can't I just compare the numbers I see in the exponents?

Because expressions like $ (x^3)^5 $ need to be simplified first! The power rule changes $ (x^3)^5 $ to $ x^{15} $, which is much larger than it initially appears.

Q: How do I remember when to add vs multiply exponents?

Add when multiplying powers: $ x^3 \times x^4 = x^{3+4} $. Multiply when raising a power to a power: $ (x^3)^5 = x^{3 \times 5} $. Think: multiplication becomes addition, power becomes multiplication.

Q: What if x is negative? Does that change the answer?

For comparing which expression is greater, we only need to compare exponents when x > 1. The expression with the highest exponent will always be greatest regardless of x's sign (assuming x ≠ 0, ±1).

Q: Why does $ \frac{x^9}{x^2} $ equal $ x^7 $ ?

Use the quotient rule: when dividing powers with the same base, subtract the exponents. So $ \frac{x^9}{x^2} = x^{9-2} = x^7 $.

Q: Can two different expressions have the same value?

Absolutely! In this problem, both $ x^3 \times x^4 $ and $ \frac{x^9}{x^2} $ simplify to $ x^7 $, so they're equal.

Exponent Rules with Power Comparisons

Which value is greater?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Select 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 When there's a power of a power, the combined power is the product of the powers

00:13 When dividing powers with equal bases

00:16 The power of the result equals the difference of the powers

00:19 We'll determine the largest power, this is the largest value

00:22 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which value is greater?

Step-by-step solution

To determine which expression has the greatest value, we apply the exponent rules to simplify each choice:

For $x^3 \times x^4$ , using the product rule: $x^3 \times x^4 = x^{3+4} = x^7$ .
For $(x^3)^5$ , using the power of a power rule: $(x^3)^5 = x^{3 \times 5} = x^{15}$ .
$x^{10}$ is already in its simplest form.
For $\frac{x^9}{x^2}$ , using the quotient rule: $\frac{x^9}{x^2} = x^{9-2} = x^7$ .

To identify the greater value, we compare the exponents:

$x^7$ from choices 1 and 4.
$x^{15}$ from choice 2.
$x^{10}$ from choice 3.

The expression with the largest exponent is $(x^3)^5$ or $x^{15}$ .

Therefore, the expression with the greatest value is $(x^3)^5$ .

Final Answer

$(x^3)^5$

Key Points to Remember

Essential concepts to master this topic

Product Rule: When multiplying powers with same base, add exponents
Power Rule: $(x^3)^5 = x^{3 \times 5} = x^{15}$ multiplies exponents
Compare: Simplify all expressions first, then compare final exponents ✓

Common Mistakes

Avoid these frequent errors

Comparing expressions without simplifying first
Don't compare $(x^3)^5$ directly to $x^{10}$ = wrong comparison! You can't tell which is greater without applying exponent rules first. Always simplify each expression completely before comparing exponents.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just compare the numbers I see in the exponents?

Because expressions like $(x^3)^5$ need to be simplified first! The power rule changes $(x^3)^5$ to $x^{15}$ , which is much larger than it initially appears.

How do I remember when to add vs multiply exponents?

Add when multiplying powers: $x^3 \times x^4 = x^{3+4}$ . Multiply when raising a power to a power: $(x^3)^5 = x^{3 \times 5}$ . Think: multiplication becomes addition, power becomes multiplication.

What if x is negative? Does that change the answer?

For comparing which expression is greater, we only need to compare exponents when x > 1. The expression with the highest exponent will always be greatest regardless of x's sign (assuming x ≠ 0, ±1).

Why does $\frac{x^9}{x^2}$ equal $x^7$ ?

Use the quotient rule: when dividing powers with the same base, subtract the exponents. So $\frac{x^9}{x^2} = x^{9-2} = x^7$ .

Can two different expressions have the same value?

Absolutely! In this problem, both $x^3 \times x^4$ and $\frac{x^9}{x^2}$ simplify to $x^7$ , so they're equal.

What's the fastest way to solve this type of problem?

Follow these steps: 1) Simplify each expression using exponent rules, 2) Write them all with single exponents, 3) Pick the expression with the highest exponent!

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Mathematical Comparison: Determining the Greater Value Between Two Numbers

Exponent Rules with Power Comparisons

❤️ 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 can't I just compare the numbers I see in the exponents?

How do I remember when to add vs multiply exponents?

What if x is negative? Does that change the answer?

Why does x9x2 \frac{x^9}{x^2} x2x9​ equal x7 x^7 x7?

Can two different expressions have the same value?

What's the fastest way to solve this type of problem?

🌟 Unlock Your Math Potential

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