Basic Squaring / Square Roots - SAT Math

Let us factor 108 and 81

Notice how all of the quantities in square roots are divisible by 9

Simplifying, this becomes

In order to solve for , first note that all of the square root terms on the left side of the equation have a common factor of 9 and 9 is a perfect square:

Simplifying, this becomes:

Note that all of the square root terms share a common factor of 36, which itself is a square of 6:

Factoring from both terms on the left side of the equation:

Note that both and have a common factor of and is a perfect square:

From here, we can factor out of both terms on the lefthand side

To begin with, factor out the contents of the radicals. This will make answering much easier:

They both have a common factor . This means that you could rewrite your equation like this:

This is the same as:

These have a common . Therefore, factor that out:

These three roots all have a in common; therefore, you can rewrite them:

Now, this could be rewritten:

Now, note that

Therefore, you can simplify again:

Now, that looks messy! Still, if you look carefully, you see that all of your factors have ; therefore, factor that out:

This is the same as:

Examining the terms underneath the radicals, we find that and have a common factor of . itself is a perfect square, being the product of and . Hence, we recognize that the radicals can be re-written in the following manner:

, and .

The equation can then be expressed in terms of these factored radicals as shown:

Factoring the common term from the lefthand side of this equation yields

Divide both sides by the expression in the parentheses:

Divide both sides by to yield by itself on the lefthand side:

Simplify the fraction on the righthand side by dividing the numerator and denominator by :

This is the solution for the unknown variable that we have been required to find.

To simplify, break down each square root into its component factors:

You can remove pairs of factors and bring them outside the square root sign. At this point, since each term shares , you can add them together to yield the final answer:

Take each fraction separately first:

(2√3)/(√2) = \[(2√3)/(√2)\] * \[(√2)/(√2)\] = (2 * √3 * √2)/(√2 * √2) = (2 * √6)/2 = √6

Similarly:

(4√2)/(√3) = \[(4√2)/(√3)\] * \[(√3)/(√3)\] = (4√6)/3 = (4/3)√6

Now, add them together:

√6 + (4/3)√6 = (3/3)√6 + (4/3)√6 = (7/3)√6

Square both sides:

x = (32)2 = 92 = 81

Begin by factoring out each of the radicals:

For the first two radicals, you can factor out a or :

The other root values cannot be simply broken down. Now, combine the factors with :

This is your simplest form.

Begin by getting your terms onto the left side of the equation and your numeric values onto the right side of the equation:

Next, you can combine your radicals. You do this merely by subtracting their respective coefficients:

Now, square both sides:

Solve by dividing both sides by :

√27 is the same as 3√3, while √12 is the same as 2√3.

3√3 + 2√3 = 5√3

(5√3)/(√3) = 5

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

Let's simplify this even further by factoring out a .

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

Let's simplify this even further by factoring out a .

Let us factor 108 and 81

First list all the integers between 20 and 80 that are squares of another integer:

52 = 25

62 = 36

72 = 49

82 = 64

In total, there are 61 integers from 20 to 80, inclusive. 61 – 4 = 57

The easiest way to narrow down a square root from a list is to look at the last number on the squared number – in this case 4 – and compare it to the last number of the answer.

70 * 70 will equal XXX0

71 * 71 will equal XXX1

72 * 72 will equal XXX4

73 * 73 will equal XXX9

74 * 74 will equal XXX(1)6

Therefore 72 is the answer. Check by multiplying it out.