### All HiSET: Math Resources

## Example Questions

### Example Question #21 : Numbers And Operations On Numbers

Simplify:

**Possible Answers:**

The expression is already simplified.

**Correct answer:**

To simplify a radical expression, first find the prime factorization of the radicand, which is 40 here.

Pair up like factors, then apply the Product of Radicals Property:

,

the simplest form of the radical.

### Example Question #21 : Numbers And Operations On Numbers

Simplify the sum:

**Possible Answers:**

The expression cannot be simplified further.

**Correct answer:**

The expression cannot be simplified further.

To simplify a radical expression, first find the prime factorization of the radicand. First, we will attempt simplify as follows:

Since no prime factor appears twice, the expression cannot be simplified further. The same holds for , since ; the same also holds for , since 11 is prime.

It follows that the expression is already in simplest form.

### Example Question #23 : Numbers And Operations On Numbers

Consider the expression .

To simplify this expression, it is necessary to first multiply the numerator and the denominator by:

**Possible Answers:**

**Correct answer:**

When simplifying a fraction with a denominator which is the sum or difference of an integer and a square root, it is necessary to first rationalize the denominator. This is accomplished by multiplying both halves of the fraction by the *conjugate* of the denominator—the result of changing the plus symbol to a minus symbol (or vice versa); therefore, both halves of the given expression must be multiplied by the conjugate of , which is .

is therefore the correct choice.

### Example Question #21 : Numbers And Operations On Numbers

Multiply:

**Possible Answers:**

**Correct answer:**

The two expressions comprise the sum and the difference of the same two expressions, and can be multiplied using the difference of squares formula:

The square of the square root of an expression is the expression itself, so:

### Example Question #31 : Numbers And Operations On Numbers

Simplify:

**Possible Answers:**

The expression is already simplified.

**Correct answer:**

To simplify a radical expression, first find the prime factorization of the radicand, which is 120 here.

Pair up like factors, then apply the Product of Radicals Property:

,

the simplest form of the radical.

### Example Question #31 : Numbers And Operations On Numbers

Simplify the difference:

**Possible Answers:**

The expression cannot be simplified further.

**Correct answer:**

The expression cannot be simplified further.

To simplify a radical expression, first, find the prime factorization of the radicand. First, we will attempt simplify as follows:

Since no prime factor appears twice, the expression cannot be simplified further. The same holds for , since .

It follows that the expression is already in simplest form.