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 Post subject: GMAT AlgebraPosted: Tue Aug 17, 2010 12:49 pm

Joined: Sun May 30, 2010 3:15 am
Posts: 424
If a is a positive integer then 3²ª + 36ª / 2²ª =
A. (9/4)ª + 9ª
B. 9ª + 18ª
C. 18ª
D. 0
E. 2 × 9ª

(E) 3²ª + 36ª / 2²ª = 9ª + 36ª / 4ª = 9ª + (36/4)ª = 9ª + 9ª = 2 × 9ª, choice (E).
Plugging in different values for a until all choices except one are eliminated is also an option.
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Where did the 9ª + 36ª / 4ª = 9ª + (36/4)ª = 9ª + 9ª = 2 × 9ª come from?

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Tue Aug 17, 2010 1:12 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498
Let us follow the reasoning step-by-step and I'll be explaining in brief the properties of the exponents we use along the way. An extended explanation of a property you may find at the end of this post.

Originally we have:
3²ª + 36ª / 2²ª =

Let us use the property nʷª = (nʷ)ª = (nª)ʷ, where n, w, a are positive integers.

= (3²)ª + 36ª / (2²)ª =

Then we calculate the numbers within brackets:

= (9)ª + 36ª / (4)ª =

Let us use the property nª / wª = (n / w)ª, where n, w, a are positive integers.

= 9ª + (36/4)ª =

Then we calculate the number within brackets:

= 9ª + (9)ª =

Adding two equal values is the same as multiplying it by 2, so at the end we get:

= 2 × 9ª, the right answer is choice (E).

-------------
EXTENDED EXPLANATIONS for the PROPERTIES WE USED:
n, w, a are positive integers.

1. nʷª = (nʷ)ª = (nª)ʷ
Why is it so?

The definition of an exponent tells us that:
nʷª = n × ... × n {wa times}

(nʷ)ª = (n × ... × n {w times})ª = (n × ... × n {w times}) × ... × (n × ... × n {w times}) {a brackets} = n × ... × n {wa times}

(nª)ʷ = (n × ... × n {a times})ʷ = (n × ... × n {a times}) × ... × (n × ... × n {a times}) {w brackets} = n × ... × n {aw times}

We can see that all the formulas result in the same number of multiplied n.

2. nª / wª = (n / w
Why is it so?
The definition of an exponent tells us that:
nª / wª = (n × ... × n {a times}) / (w × ... × w {a times}) = (n/w) × ... × (n/w) {a times} = (n / w

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Wed Sep 15, 2010 4:05 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424
Why isn't (36ª / 4ª) = 9? When we divide exponents I thought we subtract?

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Wed Sep 15, 2010 4:14 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
We subtract exponents when we deal with the same base, e.g. 4²ª / 4 ª = 4ª. In this case the bases differ.

Take a look at proper calculations once again:
36ª / 4ª = (36/4)ª = 9ª
or
36ª / 4ª = (4 × 9)ª / 4ª = (4ª × 9ª) / 4ª = 9ª

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Tue Nov 02, 2010 3:42 am

Joined: Sun May 30, 2010 3:15 am
Posts: 424

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Tue Nov 02, 2010 3:45 am

Joined: Sun May 30, 2010 2:23 am
Posts: 498
C. 18ª
E. 2 × 9ª

18ª = (2 × 9)ª = 2ª × 9ª > 2 × 9ª (if a > 1)

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Mon Feb 21, 2011 7:26 pm

Joined: Sun May 30, 2010 3:15 am
Posts: 424
Can you explain why 9ª + 9ª doesn't go to 2(9ª) = 18ª?

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 Post subject: Re: math (test 2, question 8): algebra, exponentsPosted: Mon Feb 21, 2011 7:31 pm

Joined: Sun May 30, 2010 2:23 am
Posts: 498
2 × 9ª = 2 × (9 × 9 × ... × 9)
18ª = (2 × 9)ª = 2ª × 9ª = (2 × 2 × ... × 2)(9 × 9 × ... × 9)

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