SAT Math : Squaring / Square Roots / Radicals

Study concepts, example questions & explanations for SAT Math

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Example Questions

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Example Question #1 : Squaring / Square Roots / Radicals

Simplify: 

Possible Answers:

Correct answer:

Explanation:

If you don't already have the pattern memorized, use FOIL. It's best to write out the parentheses twice (as below) to avoid mistakes:

Example Question #1 : Squaring / Square Roots / Radicals

Simplify the radical.

\sqrt{3283}

Possible Answers:

7\sqrt{67}

67\sqrt{49}

7\sqrt{63}

57.3

56

Correct answer:

7\sqrt{67}

Explanation:

We can break the square root down into 2 roots of 67 and 49. 49 is a perfect square and reduces to 7.

Example Question #2 : Squaring / Square Roots / Radicals

Simplify: 

Possible Answers:

Correct answer:

Explanation:

If you don't already have the pattern memorized, use FOIL. It's best to write out the parentheses twice (as below) to avoid mistakes:

Example Question #2 : Squaring / Square Roots / Radicals

x2 = 36

Quantity A: x

Quantity B: 6

Possible Answers:

Quantity A is greater

The relationship cannot be determined from the information given

Quantity B is greater

The two quantities are equal

Correct answer:

The relationship cannot be determined from the information given

Explanation:

x2 = 36 -> it is important to remember that this leads to two answers. 

x = 6 or x = -6. 

  If x = 6: A = B.

  If x = -6: A < B.

Thus the relationship cannot be determined from the information given.

Example Question #1 : Squaring / Square Roots / Radicals

According to Heron's Formula, the area of a triangle with side lengths of a, b, and c is given by the following:

Hero

where s is one-half of the triangle's perimeter. 

 

What is the area of a triangle with side lengths of 6, 10, and 12 units?

Possible Answers:

8√14

14√2

4√14

48√77

12√5

Correct answer:

8√14

Explanation:

We can use Heron's formula to find the area of the triangle. We can let a = 6, b = 10, and c = 12.

In order to find s, we need to find one half of the perimeter. The perimeter is the sum of the lengths of the sides of the triangle.

Perimeter = a + b + c = 6 + 10 + 12 = 28

In order to find s, we must multiply the perimeter by one-half, which would give us (1/2)(28), or 14.

Now that we have a, b, c, and s, we can calculate the area using Heron's formula. 

Hero

Hero2

 

 

Example Question #1 : Factoring Squares

Simplify the radical expression.

Possible Answers:

Correct answer:

Explanation:

Look for perfect cubes within each term. This will allow us to factor out of the radical.

Simplify.

Example Question #3 : How To Factor A Common Factor Out Of Squares

Simplify the expression.

Possible Answers:

Correct answer:

Explanation:

Use the distributive property for radicals. 

Multiply all terms by .

Combine terms under radicals.

Look for perfect square factors under each radical.  has a perfect square of . The can be factored out.

Since both radicals are the same, we can add them.

Example Question #30 : Powers & Roots Of Numbers

Which of the following expression is equal to

 

Possible Answers:

Correct answer:

Explanation:

When simplifying a square root, consider the factors of each of its component parts:

Combine like terms:

Remove the common factor, :

Pull the  outside of the equation as :

                       

Example Question #3 : Squaring / Square Roots / Radicals

Which of the following is equal to the following expression?

Possible Answers:

Correct answer:

Explanation:

First, break down the components of the square root:

Combine like terms. Remember, when multiplying exponents, add them together:

Factor out the common factor of :

Factor the :

Combine the factored  with the :

Now, you can pull  out from underneath the square root sign as :

Example Question #1 : Squaring / Square Roots / Radicals

Which of the following expressions is equal to the following expression?

Possible Answers:

Correct answer:

Explanation:

First, break down the component parts of the square root:

Combine like terms in a way that will let you pull some of them out from underneath the square root symbol:

Pull out the terms with even exponents and simplify:

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