ACT Math: Exponents and Roots

 

Exponents, or powers, are numbers that tell us to how many times to multiply a number by itself. For example, 2⁶ = 2 * 2 * 2 * 2 * 2 * 2. You might read this as “two to the sixth power,” and our answer would be 64. For the ACT Math, you’ll really want to know how to manipulate expressions with exponents. There are many rules that dictate how we manipulate expressions with exponents, so let’s get started.

When adding algebraic expressions that have the same bases and exponents, I can add their coefficients:
X^³+X^{³ =}=2 X^³
3X⁵ +2 X⁵ =5x⁵
Remember that you can only add the coefficients when the base and exponent are the same. Adding and subtracting will never result in a change in exponent (e.g. 2 X^³ +2 X^³ does not equal 4x⁶ )
When multiplying expressions or terms that have the same base, just add the exponents.
X^³ * X³ ⁼=X⁶
When dividing numbers or terms with the same base, subtract the exponents.
X⁷ / X^{4 =}= x³

When raising an exponent to another exponent, you must multiply the exponents. In order to raise an exponent to an exponent, you must place parentheses around the original expression and then place the exponent outside the parentheses.
(X^³ )³= X⁹
Most of the time, if students are going to mix something up about manipulating expressions with exponents, they often mix up multiplying exponents with raising an exponent to an exponent. The former involves adding exponents while the latter involves multiplying exponents.

A number raised to a negative power is equal to “1 over” (a.k.a. the reciprocal of) that base raised to the opposite (positive) of that power.
X^-³=1 / X^³
 

Exponent Reminders

Zero: Anything raised to the power of zero is 1
One: Anything raised to the power of 1 is itself.

Roots and radicals are the inverse of exponents. If three squared (3*3) is nine, then the square root of nine (what number multiplied by itself yields 9?) is three. Note that “square roots,” though the most common on the test, are not the only roots tested. The test may ask you to work with square roots, cubed roots, fourth roots, and so on. The cubed root of 8, for example, is 2, since 2*2*2 is 8.

You cannot add or subtract roots. You must work out each root and then add the numbers. So, √9 + √16 is not √25. You must convert the roots to 3 and 4, respectively, and our answer is 7.
You can multiply or divide roots only if they are of the same degree (i.e. they are both square roots, cubed roots, fourth roots, etc.). So if you want to multiply two roots of the same degree, just multiply the numbers under the root sign and place that product under a new root sign with the same degree.
√26 * √2 = √52

Sometimes on the test, you’ll see roots in the answer choices written as the product of a number and a square root. These expressions are in simplified form. Take the √52 we see above. This is not in simplified above. To simplify  it, first find all its prime factors by drawing a factor tree
52
/   \
2  26
/     \
13    2
o   Notice that the prime factors of 52 are 2, 2, and 13. We can write √52 as a product of 2 and √13. When simplifying radicals, we want to identify the pairs of prime factors; in this case, we have two 2s, so while we didn’t notice it in the first place, we can write √52 as √13*4, which we can write as √13 * √4, which simplifies to 2 √13.