Digital SAT Math
A guide to radicals and rational exponents on the digital SAT
What are radicals and rational exponents?
Exponential expressions are algebraic expressions with a coefficient, one or more variables, and one or more exponents. For example, in the expression
: is the coefficient. is the base. is the exponent.
, is multiplied by times:
An expression can also be raised to an exponent. For example, for
, the expression is multiplied by itself times:
Rational exponents refer to exponents that are/can be represented as fractions:
, , and are all considered rational exponents. Radicals are another way to write rational exponents. For example, and are equivalent.
In this lesson, we'll:
- Review the rules of exponent operations with integer exponents
- Apply the rules of exponent operations to rational exponents
- Make connections between equivalent rational and radical expressions
You can learn anything. Let's do this!
What are the rules of exponent operations?
Powers of products & quotients (integer exponents)
The rules of exponent operations
Adding and subtracting exponential expressions
When adding and subtracting exponential expressions, we're essentially combining like terms. That means we can only combine exponential expressions with both the same base and the same exponent.
Multiplying and dividing exponential expressions
When multiplying two exponential expressions with the same base, we keep the base the same, multiply the coefficients, and add the exponents. Similarly, when dividing two exponential expressions with the same base, we keep the base the same and subtract the exponents.
When multiplying or dividing exponential expressions with the same exponent but different bases, we multiply or divide the bases and keep the exponents the same.
Raising an exponential expression to an exponent and change of base
When raising an exponential expression to an exponent, raise the coefficient of the expression to the exponent, keep the base the same, and multiply the two exponents.
When the bases are numbers, we can use a similar rule to change the base of an exponential expression.
This is useful for questions with multiple terms that need to be written in the same base.
A base raised to a negative exponent is equivalent to
divided by the base raised to the of the exponent.
A nonzero base raised to an exponent of
is equal to .
How do the rules of exponent operations apply to rational exponents?
Every rule that applies to integer exponents also applies to rational exponents.
try: divide two rational expressions
In order to divide
by , we .
the coefficients and
the exponents of
Try: raise an exponential expression to an exponent
, we and .
How are radicals and fractional exponents related?
Rewriting roots as rational exponents
Roots and rational exponents
Squares and square roots are inverse operations: they "undo" each other. For example, if we take the square root of
squared, we get .
The reason for this becomes more apparent when we rewrite square root as a fractional exponent:
, and .
When rewriting a radical expression as a fractional exponent, any exponent under the radical symbol (
) becomes the numerator of the fractional exponent, and the value to the left of the radical symbol (e.g., ) becomes the denominator of the fractional exponent. Square root is equivalent to .
All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents. Similarly, for radical expressions:
When working with radical expressions with the same radical, we can choose whether to convert to fractional exponents first or multiply what's under the radical symbol first to our advantage.
Try: determine equivalent expressions
Determine whether each of the radical expressions below is equivalent to
Practice: multiply rational expressions
Which of the following is equivalent to
Practice: change bases
for positive integers and , what is one possible value of ?
Practice: raise to a negative exponent
, where , what is in terms of ?
Practice: simplify radical expressions
Which of the following is equivalent to the expression above?
Things to remember
Adding and subtracting exponential expressions:
Multiplying and dividing exponential expressions:
Raising an exponential expression to an exponent and change of base:
All of the rules that apply to exponential expressions with integer exponents also apply to exponential expressions with fractional exponents.
Want to join the conversation?
- I feel like giving up(42 votes)
- dumb ways to die(37 votes)
- these radicals are so rad(13 votes)
- my DSAT is in august any tips ? please help your sister out its my first time taking any sat and im rlly nervous. i took a practice test and got 1050 any tips?(10 votes)
- So, for English try to read the questions first, then when you look at the text you already know what they are asking for and it will be easier to find the answer, In math, try to memorize the concepts, and the question patterns, cause they repeat questions but with different wording and numbers, and DONT let any answer in blank, try to answer all the questions even though you do not know the answers, it helps cause you have more chance to get more points(10 votes)
- Pls if you know any shortcut to any math problem please share it with us:)(9 votes)
- We call this lesson at our school surds and indices. It's pretty much the same. I've researched it. It's just a matter of naming. So for anyone here who learnt this lesson as surds and indices don't worry the content is the exact same.(9 votes)