# HiSET: Math : Irrational numbers

## Example Questions

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### Example Question #1 : Irrational Numbers

Simplify the expression

Explanation:

An expression with a radical expression in the denominator is not simplified, so to simplify, it is necessary to rationalize the denominator. This is accomplished by multiplying both numerator and denominator by the given square root, , as follows:

can be simplified by taking the prime factorization of 12, and taking advantage of the Product of Radicals Property.

, so

Returning to the original expression and substituting:

,

the correct response.

### Example Question #1 : Properties Of Operations With Real Numbers

Consider the expression .

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

Explanation:

When simplifying an fraction with a denominator which is the product of an integer and a square root expression, it is necessary to first rationalize the denominator. This is accomplished by multiplying both halves of the fraction by the square root expression. The correct response is therefore .

### Example Question #1 : Properties Of Operations With Real 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 second expression can be rewritten by the Power of a Product Property:

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

By order of operations, multiply, then subtract:

### Example Question #11 : 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:

By distribution:

### Example Question #1 : Properties Of Operations With Real Numbers

Multiply:

None of the other choices gives the correct response.

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 #1 : Properties Of Operations With Real Numbers

Simplify the expression:

Explanation:

An expression with a radical expression in the denominator is not simplified, so to simplify, it is necessary to rationalize the denominator. This is accomplished by multiplying both numerator and denominator by the given square root, , as follows:

The expression can be simplified further by dividing the numbers outside the radical by greatest common factor 5:

This is the correct response.

### Example Question #1 : Properties Of Operations With Real Numbers

Simplify:

Explanation:

To simplify a radical expression, first find the prime factorization of the radicand, which is 66 here.

There are no repeated prime factors, so the expression is already in simplified form.

### Example Question #1 : Irrational Numbers

Simplify the sum:

The expression cannot be simplified further.

Explanation:

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

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

By similar reasoning,

11 is a prime number, so  cannot be simplified. We can replace it with .

Through substitution, and the distribution property:

### 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 32 here.

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

,

the simplest form of the radical.

### Example Question #1 : Irrational Numbers

Simplify the difference:

None of the other choices gives the correct response.

Explanation:

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

By the Product of Radicals Property,

11 is a prime number, so  cannot be simplified. We can replace it with .

Through substitution, and the distribution property:

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