All ACT Math Resources
Example Questions
Example Question #8 : How To Multiply Complex Numbers
Which of the following is equal to ?
Remember that since , you know that is . Therefore, is or . This makes our question very easy.
is the same as or
Thus, we know that is the same as or .
Example Question #41 : Squaring / Square Roots / Radicals
Complex numbers take the form , where is the real term in the complex number and is the nonreal (imaginary) term in the complex number.
Simplify the following expression, leaving no complex numbers in the denominator.
Solving this problem requires eliminating the nonreal term of the denominator. Our best bet for this is to cancel the nonreal term out by using the conjugate of the denominator.
Remember that for all binomials , there exists a conjugate such that .
This can also be applied to complex conjugates, which will eliminate the nonreal portion entirely (since )!
Multiply both terms by the denominator's conjugate.
Simplify. Note .
FOIL the numerator.
Combine and simplify.
Simplify the fraction.
Thus, .
Example Question #1 : Exponents And Rational Numbers
Which real number satisfies ?
Simplying the equation, we get .
This further simplifies to .
satisfies this equation. You could also use to determine that .
Example Question #1 : Exponents And Rational Numbers
Find the value of if
When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent. Therefore, the equation can be rewritten as . From here we can proceed one of two ways. We can either solve x for or . Let's solve the first equation. We simply multiply 4 by itself until we reach a value of 64. , ,, and so on. Since , we know that x = 3.
We can repeat this process for the second equation to get , confirming our previous answer. However, since the ACT is a timed test, it is best to only solve one of the equations and move on. Then, if you have time left once all of the questions have been answered, you can come back and double check your answer by solving the other equation.
Example Question #1 : Exponents And Rational Numbers
If and , what is ?
Start from the inside. . Then, .
Example Question #1 : Exponents And Rational Numbers
Simplify by expressing each term in exponential form.
None of these are correct.
The rule for exponential ratios is .
Using this, we can convert the numerator and denominator quickly.
The middle step in the coversion is important, as there is a big difference between and , and likewise for the denominator.
Next, we can further simplify by remembering that
.
Find the least common denominator and simplify:
Thus,
Example Question #1 : Exponents And Rational Numbers
Often, solving a root equation is as simple as switching to exponential form.
Simplify into exponential form:
None of these are correct.
The rule for exponential ratios is .
Using this, we can convert the numerator and denominator quickly.
Next, we can further simplify by remembering that
.
Find the least common denominator and simplify:
Thus, our answer is . (Remember, the problem asked for exponential form!)
Example Question #1 : Exponents And Rational Numbers
Which of the following is a value of that satisfies ?
When you have a logarithm in the form
,
it is equal to
.
Using the information given, we can rewrite the given equation in the second form to get
.
Now solving for we get the result.
Example Question #1 : Exponents And Rational Numbers
Solve for :
When you have a logarithm in the form
,
it is equal to
.
We can rewrite the given equation as
Solving for , we get
.
Example Question #2 : Exponents And Rational Numbers
Solve for :
When you have a logarithm in the form
,
it is equal to
.
We can rewrite the given equation as
Solving for , we get
.