Comparing Values with Condition a > 1: Finding the Maximum

Q: Why does a bigger exponent mean a bigger value when a > 1?

When a > 1, multiplying by a makes numbers larger. So $ a^9 = a \times a \times a \times a \times a \times a \times a \times a \times a $ is much bigger than $ a^5 = a \times a \times a \times a \times a $.

Q: What happens with negative exponents like $ a^{-3} $ ?

Negative exponents mean fractions: $ a^{-3} = \frac{1}{a^3} $. Since a > 1, we get $ \frac{1}{a^3} , which is smaller than positive powers like \( a^9 $.

Q: How do I subtract exponents when one is negative?

Remember: subtracting a negative is the same as adding! So $ \frac{a^3}{a^{-2}} = a^{3-(-2)} = a^{3+2} = a^5 $. Always be careful with the signs!

Q: Can I just plug in a = 2 to check my answer?

Yes! That's a great checking method. With a = 2: $ a^9 = 2^9 = 512 $, $ a^5 = 2^5 = 32 $, $ a^{-2} = 2^{-2} = 0.25 $, $ a^{-3} = 2^{-3} = 0.125 $. Clearly 512 is largest!

Q: What if a was between 0 and 1 instead?

Great question! If 0 , then higher exponents would actually give smaller values. But since the problem states a > 1, we know higher positive exponents mean larger values.

Exponent Rules with Positive Bases

Which value is the largest?

given that $a>1$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Identify the largest value

00:03 When dividing powers with equal bases

00:07 The power of the result equals the difference between the powers

00:11 We'll apply this formula to our exercise and subtract the powers

00:16 We'll use this method to solve all sections

00:41 Identify the largest power

00:50 Number ( A )is greater than 1 according to the given data

00:59 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which value is the largest?

given that $a>1$

Step-by-step solution

Note that in all options there are fractions where both numerator and denominator have terms with identical bases, therefore we will use the division law between terms with identical bases to solve the exercise:

$\frac{c^m}{c^n}=c^{m-n}$ Let's apply it to the problem, first we'll simplify each of the suggested options using the above law (options in order):

$\frac{a^{17}}{a^{20}}=a^{17-20}=a^{-3}$ $\frac{a^{10}}{a^1}=a^{10-1}=a^9$ $\frac{a^3}{a^{-2}}=a^{3-(-2)}=a^{3+2}=a^5$ $\frac{a^2}{a^4}=a^{2-4}=a^{-2}$ Let's return to the problem, given that:

$a>1$ therefore the option with the largest value will be the one where $a$ has the largest exponent (for emphasis - a positive exponent is greater than a negative exponent),

meaning the option:

$a^9$ above is correct, it came from option B in the answers,

therefore answer B is correct.

Final Answer

$\frac{a^{10}}{a^1}$

Key Points to Remember

Essential concepts to master this topic

Division Rule: When dividing powers with same base, subtract exponents
Technique: $\frac{a^{10}}{a^1} = a^{10-1} = a^9$ gives largest positive exponent
Check: Since a > 1, higher exponents mean larger values: $a^9 > a^5 > a^{-2} > a^{-3}$ ✓

Common Mistakes

Avoid these frequent errors

Comparing original fractions without simplifying first
Don't try to compare $\frac{a^{10}}{a^1}$ vs $\frac{a^3}{a^{-2}}$ directly = confusing comparison! The fractions look complex and it's hard to see which is bigger. Always simplify using $\frac{a^m}{a^n} = a^{m-n}$ first to get clear exponents like $a^9$ vs $a^5$ .

Practice Quiz

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

FAQ

Everything you need to know about this question

Why does a bigger exponent mean a bigger value when a > 1?

When a > 1, multiplying by a makes numbers larger. So $a^9 = a \times a \times a \times a \times a \times a \times a \times a \times a$ is much bigger than $a^5 = a \times a \times a \times a \times a$ .

What happens with negative exponents like $a^{-3}$ ?

Negative exponents mean fractions: $a^{-3} = \frac{1}{a^3}$ . Since a > 1, we get $\frac{1}{a^3} < 1$ , which is smaller than positive powers like $a^9$ .

How do I subtract exponents when one is negative?

Remember: subtracting a negative is the same as adding! So $\frac{a^3}{a^{-2}} = a^{3-(-2)} = a^{3+2} = a^5$ . Always be careful with the signs!

Can I just plug in a = 2 to check my answer?

Yes! That's a great checking method. With a = 2: $a^9 = 2^9 = 512$ , $a^5 = 2^5 = 32$ , $a^{-2} = 2^{-2} = 0.25$ , $a^{-3} = 2^{-3} = 0.125$ . Clearly 512 is largest!

What if a was between 0 and 1 instead?

Great question! If 0 < a < 1, then higher exponents would actually give smaller values. But since the problem states a > 1, we know higher positive exponents mean larger values.

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Comparing Values with Condition a > 1: Finding the Maximum

Exponent Rules with Positive Bases

❤️ 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 does a bigger exponent mean a bigger value when a > 1?

What happens with negative exponents like $a^{-3}$ ?

How do I subtract exponents when one is negative?

Can I just plug in a = 2 to check my answer?

What if a was between 0 and 1 instead?

🌟 Unlock Your Math Potential

More Questions

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Finding the Largest Value: Comparing Numbers with Constraint a>1

Finding the Largest Value When a > 1: Inequality Comparison

Value Comparison with Variable a: Finding the Largest When a>1

Simplify the Expression: 12b⁴ ÷ 4b⁻⁵ Using Power Rules

Comparing Values with Condition a > 1: Finding the Maximum

Exponent Rules with Positive Bases

❤️ 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 does a bigger exponent mean a bigger value when a > 1?

What happens with negative exponents like a−3 a^{-3} a−3?

How do I subtract exponents when one is negative?

Can I just plug in a = 2 to check my answer?

What if a was between 0 and 1 instead?

🌟 Unlock Your Math Potential

More Questions

Power of a Quotient Rule for Exponents

Finding the Largest Value: Comparing Numbers with Constraint a>1

Finding the Largest Value When a > 1: Inequality Comparison

Value Comparison with Variable a: Finding the Largest When a>1

Simplify the Expression: 12b⁴ ÷ 4b⁻⁵ Using Power Rules

What happens with negative exponents like $a^{-3}$ ?