# HiSET: Math : Irrational numbers

## Example Questions

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

Simplify:

Explanation:

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:

The expression cannot be simplified further.

The expression cannot be simplified further.

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

The expression cannot be simplified further.

The expression cannot be simplified further.

Explanation:

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.

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