GRE Quantitative: Algebra – Rules of Exponents

Algebra and exponents have always been one of the toughest sections of the GRE syllabus in the Quantitative Reasoning division. Although addition has always been sweet, even multiplication was cool. But are you adept at new operations that grows numbers at a faster rate than addition or multiplication? Want to explain how many things in this universe grow and shrink. You have your answer and that operation is known as exponents.

Let’s understand what exponents are:

To understand exponents in greater depth, let us start it with a game. Grab a normal piece of paper and start folding it in half. Scientifically, it’s said that you can fold a paper in half, a maximum of eight times and not more than that. But in the end, you will notice that the paper after folding for seven to eight times is considerably much thicker than the piece of paper you initially took.

Now, take a paper which can hypothetically be folded about 50 times. How much do you consider its thickness will be? 5 inches? 12 inches? Keep guessing, but even your most ambitious guesses won’t be right. The correct answer to this question is the distance between the sun and the earth. Amazed?

Let’s understand how.

The thickness of any usual bit of paper is around 1/200th of an inch. Hence, every time if you fold that paper, you will be doubling its thickness. So, initially you will be having the original thickness of the paper multiplied times 2, times 2, times 2, …….etc. So eventually you have multiplied the original thickness of the paper 2⁵⁰ times. The number 2⁵⁰ is extraordinarily an enormous number that even if it is multiplied by a number as small as 1/200, it won’t make a huge difference.

In this game, the value 2⁵⁰ is the exponent value, where 2 is base and 50 is the exponent. The exponent is used for representing the number of times the base value is multiplied with itself. Both the base and exponent can be any real number; it is not only confined to positive integers.

Properties:

• An even exponent will always yield positive values.
• Zero raised to any power will always yield zero.
• 1 raised to any power will yield 1.
• -1 raised to even power yields 1, otherwise it’s -1.
• When a proper fraction is raised to any power, the resultant is less than the actual number.
• When a fraction is raised to a power, the exponent value contributes to both, numerator and the denominator.
• Multiplying exponents with same base: If the base is same then simply add the exponents.
• Dividing exponents with same base: Simply subtract the exponents and keep the base same.
• When you are raising an exponential power to a power then to get the final value multiply the exponents.
• When multiplying exponents with different bases but same exponents, then keep the exponent same and multiply the bases together.
• When dividing exponents with different bases but same exponents, then keep the exponent same and divide the bases.
• For negative exponents, flip the base. That is, if it is in numerator make it in denominator or vice versa.

These are some of the properties that can help you solve exponents with extreme ease in your GRE exam.

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