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Exponents: an Important Topic on the SAT Math Section

Some of the most confusing questions on the SAT math section are based on exponents. This is because intimidating questions can be made with exponents, which can be quickly solved by a student who knows them well.


The topic of exponents is an important one on the SAT, and exponents and square roots appear frequently. (See important topics on the SAT). Today we'll look at some of their fundamentals, as well as at common mistakes students make when dealing with exponents.


Today's article will serve as a jumping board for one of the question types we'll discuss later, complex exponent questions on the SAT.


What are exponents?

Exponents are values which show how many times to multiply a base. A base is the number being multiplied by itself.


Exponents are also known as powers. However, the word "power" can also be used to refer to the whole package, base plus exponent together.


Let's look at the following power and equation:



In the above equation, y is the base and 4 is the exponent. The exponent is sometimes known as the power, but "power" can also refer to the package of y to the 4th. For the purpose of clarity, that's how we'll use it in this article.


Exponent fundamentals

There are several important rules to remember when dealing with exponents. We'll do a brief review of each one, so that later we can move on to more complicated questions.


Product of powers rule

When multiplying like bases with respective exponents, keep the base and add the exponents.

That is:


Let's consider, for example, the following expression:



The final number is too large to be calculated on a calculator, but the phrase can be simplified. Note that the two things being multiplied have a common base. This is crucial, and the basis of a common mistake in regard to exponent calculations:


When the bases being multiplied are different, and their exponents are something other than 0 or 1, the expression can't be simplified.


Here, however, the base is the same. Following our rule, we'll keep the same base (4) and add the exponents. 17 + 5 = 22, hence:



Let's try another one. Let's look at the following expression:



Here, too, the base is the same. x + x = 2x, so the above expression should equal:



Let's look at one more example.



The base is the same, so we'll add the exponents.


x² + 3y - x² = 3y ⇒


This last example is reminiscent of some of the more complicated exponent questions we'll see on the SAT.


Let's move on to the next rule.


Quotient of powers rule When dividing like bases with respective exponents, keep the base and subtract the exponents.

That is:


As you may see, the quotient of powers rule is the exact counterpart of the product of powers rule. Note that the exponent of the denominator - in the equation above, y - is being subtracted from the exponent in the numerator, x. That is, the final exponent is x - y, not y - x.

Students sometimes get this confused.


Let's look at a few examples:


The base is the same - always important to check - so we'll subtract the exponents. 19 - 2 = 17, therefore:


Note that this is a large number. In the answer options it would likely be presented in this form (exponential) , rather than as a developed number.

Let's try one more example, this one with variables:



We'll subtract the two exponents. Note that the bottom exponent is negative, so be careful with parentheses:


2x - 4y - (-7y) = 2x - 4y + 7y = 2x + 3y


So:


Power of a power rule

When a number with an exponent is raised to another exponent, keep the base and multiply the exponents.

That is:


Let's see if we can understand the implications implicit in this, because this rule is a critical stepping stone in the solving of exponent questions on the SAT. Imagine that we're interested in the simplication of the following number expressed in exponent form:


The crucial point here is that 8 itself can be expressed as the power of another base:


In that case,


Following the power of a power rule,


Whenever you see a number which can be expressed as a power of a smaller base, you can use the power of a power rule. On the SAT itself, this is almost an axiom:, when you can use this rule, you should use the rule.


When you can use the power of a power rule, you should use the power of a power rule.


The power of a power rule is frequently used in the opposite way as well, from right to left. Let's look at an algebraic question which works the opposite way:


In the equation above, we're moving from a smaller base to a larger base. It depends on what the question requires.


Power of a product rule

When raising the product of two or more variables by a power, that power can be distributed to each variable. That is: