ACT Math - How to find patterns in exponents

What is the value of the expression $m^{2}+3m$ , given that $m^{\frac{3}{2}}=64$ ?

Explanation

First, calculate the value of m by taking the two-thirds root of both m3/2 and 64, leaving m=16. Then solve the expression by plugging in 16 for m. This gives 162 + 3(16) = 256 + 48 = 304.

If $x^7y^8z^{10} < 0$ , then which of the following must also be true?

Explanation

We know that the expression must be negative. Therefore one or all of the terms x7, y8 and z10 must be negative; however, even powers always produce positive numbers, so y8 and z10 will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x7 must be negative, so x must be negative. Thus, the answer is x < 0.

Evaluate $(2(x^{3})^{2})^{2}$

$2x^{12}$

$x^{7}$

$9x^{18}$

$4x^{12}$

$2x^{6}$

Explanation

$2^{2}\ast(x^{3})(x^{3})(x^{3})(x^{3})=4x^{12}$

Which of the following is a multiple of ?

Explanation

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

Now, in order for you to have a number that is a multiple of this, you will need to have at least in the prime factorization of the given number. For each of the answer choices, you have:

; This is the answer.

A professional football player’s contract states that he will earn a salary of $1 million his first year. He would then have a 15% increase every year thereafter for the next 5 years. What would he make in his 6th and final season on the contract?

$$1.75\ million$

$$2.01\ million$

$$2.32\ million$

$$3.01\ million$

$$5.75\ million$

Explanation

We can represent this as an exponential equation (just use million as a label and not a variable):

$1_m_ * (1.15)5 = $2.01_m_

For which of the following values of n will (**–**3)n represent a real number between 0 and 1?

**–**2

**–**1

Explanation

In order to transform a negative integer to a positive one, it must be taken to an even power, and to make it a fraction between 0 and 1, it must be taken to a negative power. The only choice that fits both criteria is -2.

Simplify the following:

$\frac{48^5^0+80^3^0}{4^2^0}$

Explanation

With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

Continuing the simplification:

Now, these factors have in common a . Factor this out:

Denominator

This is much simpler:

Now, return to your fraction:

$\frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}$

Cancel out the common factors of :

$\frac{2^1^2^0(2^8^0*3^5^0+5^3^0)}{2^4^0}=2^8^0(2^8^0*3^5^0+5^3^0)=2^1^6^0*3^5^0+2^8^0*5^3^0$

Simplify the following:

$\frac{25^4^4-5^8^6}{12}$

$\frac{25^2}{24}$

$\frac{4}{5^2}$

Explanation

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the into its prime factors: