Understanding the Conversion of Expressions in Mathematics

When tackling the Graduate Management Admission Test (GMAT), math can sometimes feel like you’re playing a game of chess with numbers. One question that often pops up is how to convert expressions into the format ( A \times 10^{b} ). So, let's break it down, shall we?

To convert the expression ( 25^{7} \times 4^{8} ), we first need to reflect on what these numbers represent in their most basic forms. You see, 25 and 4 can both be rewritten in terms of their prime factorization.

Let’s get to it! Did you know that 25 is equivalent to ( 5^2 )? This means, when we raise it to the power of 7, we essentially multiply the exponents:

[ 25^{7} = (5^{2})^{7} = 5^{14} ]

Pretty neat, right? Now, let’s turn our attention to the number 4. This one’s a bit clearer as it translates to ( 2^2 ). When we raise this to the power of 8, we find:

[ 4^{8} = (2^{2})^{8} = 2^{16} ]

With these conversions in hand, our original expression can now be rewritten as:

[ 25^{7} \times 4^{8} = 5^{14} \times 2^{16} ]

But here’s the kicker! To express this in the form ( A \times 10^{b} ), we need to recall that ( 10 ) is just ( 2 \times 5 ). Utilizing this identity allows us to reformat our newly derived expression.

Now, imagine you’re approaching the GMAT. You see this question pop up, and a lightbulb flicks in your mind as you realize you have the tools to tackle it! We can factor out ( 10^{14} ), leaving us with:

[ 5^{14} \times 2^{16} = 10^{14} \times 2^{2} ]

But that's not the answer we have in the choices. Why? Because we're aiming for ( A ) = ( 2^{2} \times 5^{14} ). Don't overlook these little details; they make all the difference!

So, here’s what we find as a neat little summary:

The final form we arrive at is:

[ 5^{14} \times 2^{16} = 5^{14} \times 2^{2} \times 10^{12} ]

Here’s where remembering your basic exponent rules comes into play. The answer to our conversion question is that the expression simplifies beautifully to ( 5^{14} \times 2^{16} ).

Keeping these conversions and factorizations top of mind will prepare you not just for GMAT questions but also for a lifetime of mathematical puzzles. You know what? Math is like a clever detective; every detail counts, and every expression has a story to tell. So next time you sit for a GMAT practice test, approach those numbers with confidence knowing you’ve got this!

In conclusion, mastering conversion from ( 25^{7} \times 4^{8} ) to the format ( A \times 10^{b} ) isn't merely an exercise in math; think of it more as a stepping stone to mastering a skill that will benefit you for years to come. So let’s tackle that GMAT together, one expression at a time!