The trick to answer this question is to identify key words and convert them into mathematical expressions.

Suppose a variable  and  which represents numbers.  If a number is twice the value of the second number, then either:

 or 

The "equal" means , and proceed to the right side of the equation.

Rewrite the mathematical expression for "the sum of both numbers quantity squared".

There are 2 combinations for this question.  Write the equations.

Either of these answers are correct.

Start by setting up an equation that best describes this scenario.  Bob starts with 6 candies initially.  Each lap, , Bob is awarded 3 candies.  Write this expression.

Bob also eats 7 candies after every 2 laps.  This means that Bob will have eaten an average of 3.5 candies per lap.  Continue the expression in terms of .

Since we want to know how many laps are ran before Bob will be out of candies, set the expression to zero and solve for .

A very important word in the problem is inclusive. It tells you what type of inequality sign you need. When it says inclusive, it means to include the number, which means T can be less/greater than or equal to the number. Therefore, we need to use the < sign.

Next, we need to see what kind of numbers we're dealing with in our inequality. You always have the variable less than the greater number, though not necessarily the bigger number. The numbers we have are -361 and -297. At first glance, the -361 may appear greater. After all, 361 > 297. However, because they are negative numbers, the smaller number is actually greater, meaning -361 < -297. 

Taking both of those things into account, we see the answer is 

The number of elements in a matrix is equal to the number of rows times the number of columns (the area). Given that you cannot have a non-whole number length (something like 3.8 rows), the number of columns and rows must be factors for the number of elements.

With this in mind, to find the answer to the problem, you just find which number is not one of the factors of the number of elements (48). 

• 3 is a factor: 3 * 16
• 12 is a factor: 12 * 4
• 48 is a factor: 48 * 1
• 24 is a factor: 24 * 2

The only number that isn't a factor of 48 is 18.

To solve the given the matrices

you set the variables equal to their corresponding values (same row,column) in the other matrix, as follows.

Once you isolate each variable, solve like normal.

Remember that the solution to a square root is both positive and negative.

Simple algebra.

Simple algebra.

Therefore, the values of the variables are 

In math, i is equal to the square root of -1. You can do most regular mathematical operations with it.

To simplify the expressions, you would do the following:

The Quadratic Formula is

where a, b, and c are from the quadratic equation

.

For our problem, the values of a, b, and c are 

.

Now, we just plug in the values for a, b, and c, and solve the equation.

When simplified, the answers are 

The equation for the volume of a sphere is 

where V is volume and r is the radius. If we want to find the radius of a sphere when we know its volume, we just rearrange the equation so it equals r.

From there, we just plug in the value for V we have.

When simplifying numbers with different roots, a good place to start is to try to find any whole numbers thath have powers equal to the factors of the inner number. Here are the first 4 numbers of the x⁴ series:

By looking at these numbers, we can see the 3⁴ is a factor of 810, 81 * 10. This lets us break up the top number to . Remember, you can "take out" parts of roots that have multiplication like so:

Now our equation looks like this:

Once again, we can split up the root.

There, we have our answer.

When an expression has an irrational number in the denominator, convention calls for it to be rationalized, or changed into a rational number. The way to do this, if it's a root of some sort, is multiply the numerator and denominator by the irrational number until the denominator is rational, like this:

Now, the denominator is rationalized.

To apply this to the problem, we multiply the denominator by itself 2 times (because it's a cube root) to rationalize it.

The denominator is now rationalized.