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

Question

Determine which of the following values is the largest:

Given that a>1 .

Video Solution

Solution Steps

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 exponents
00:11 We'll apply this formula to our exercise and subtract the exponents
00:17 We'll use this method to solve all sections
00:57 A number with a negative exponent turns the numerator into the denominator
01:02 We'll identify the largest exponent
01:16 This is the solution

Step-by-Step Solution

We can observe in all the given options that there are fractions where both the numerator and the denominator have terms with identical bases. Therefore we will apply the division law between terms with identical bases in order to solve the exercise:

cmcn=cmn \frac{c^m}{c^n}=c^{m-n}

Proceed to apply it to the given problem. Begin by simplifying each of the suggested options using the above law (options in order):

a4a4=a4(4)=a4+4=a8 \frac{a^4}{a^{-4}}=a^{4-(-4)}=a^{4+4}=a^8 a10a9=a109=a1=a \frac{a^{10}}{a^9}=a^{10-9}=a^1=a a4a1=a4(1)=a4+1=a5 \frac{a^4}{a^{-1}}=a^{4-(-1)}=a^{4+1}=a^5 a6a7=a67=a1 \frac{a^6}{a^7}=a^{6-7}=a^{-1}

Remember that any number when raised to the power of 1 equals the number itself, as shown below:

b1=b b^1=b

Returning to our problem, given that:

a>1

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

Meaning that option:a8 a^8 is correct,

Therefore answer A is correct.

Answer

a4a4 \frac{a^4}{a^{-4}}