HiSET - Irrational numbers

Simplify:

$\sqrt{120}$

$2\sqrt{30}$

The expression is already simplified.

$3\sqrt{20}$

$6\sqrt{10}$

$5\sqrt{12}$

Explanation

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

$120 = 2 \times 2 \times 2 \times 3 \times 5$

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

$\sqrt{120} =\sqrt{ 2 \times 2 \times 2 \times 3 \times 5} =\sqrt{ 2 \times 2 } \times \sqrt{ 2 \times 3 \times 5} = 2 \sqrt{30}$ ,

the simplest form of the radical.

Simplify:

$\sqrt{40}$

$4\sqrt{5}$

The expression is already simplified.

$2\sqrt{10}$

$5\sqrt{4}$

$5\sqrt{2}$

Explanation

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

$40 = 2 \times 2 \times 2 \times 5$

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

$\sqrt{40 }= \sqrt{2 \times 2 \times 2 \times 5} = \sqrt{2 \times 2 } \times \sqrt{ 2 \times 5} = 2 \sqrt{10}$ ,

the simplest form of the radical.

Simplify:

$\sqrt{32}$

$4\sqrt{2}$

The expression is already simplified.

$8\sqrt{2}$

$2\sqrt{8}$

$4\sqrt{8}$

Explanation

To simplify a radical expression, first find the prime factorization of the radicand, which is 32 here.

$32 = 2 \times 2 \times 2 \times 2 \times 2$

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

$\sqrt{32} = \sqrt{2 \times 2} \times\sqrt{ 2 \times 2} \times \sqrt{2 }= 2 \times 2 \times \sqrt{2 } = 4 \sqrt{2 }$ ,

the simplest form of the radical.

Simplify the expression:

$\frac{10}{\sqrt{15}}$

$\frac{2\sqrt{15}}{3}$

$\sqrt{\frac{20}{3}}$

$\frac{\sqrt{60}}{3}$

$\frac{10\sqrt{15}}{15}$

$\sqrt{\frac{60}{9}}$

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, $\sqrt{15}$ , as follows:

$\frac{10}{\sqrt{15}} = \frac{10 \times \sqrt{15}}{\sqrt{15} \times \sqrt{15}} = \frac{10 \sqrt{15}}{15}$

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

$\frac{10 \sqrt{15}}{15} = \frac{(10 \div 5) \sqrt{15}}{15 \div 5} = \frac{2 \sqrt{15}}{3}$

This is the correct response.

Simplify the difference:

$\sqrt{176} - \sqrt{11}$

$3\sqrt{11}$

$4\sqrt{11}$

$15\sqrt{11}$

$16\sqrt{11}$

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 $\sqrt{176}$ as follows:

$176 = 2 \times 2 \times 2 \times 2 \times 11$

By the Product of Radicals Property,

$\sqrt{176 }$

$= \sqrt{ 2 \times 2 \times 2 \times 2 \times 11}$

$= \sqrt{2 \times 2} \times \sqrt{2 \times 2} \times \sqrt{11 }$

$=2 \times 2 \times \sqrt{11 }$

$= 4 \sqrt{11}$

11 is a prime number, so $\sqrt{11}$ cannot be simplified. We can replace it with $1\sqrt{11}$ .

Through substitution, and the distribution property:

$\sqrt{176} - \sqrt{11}$

$= 4 \sqrt{11} -1 \sqrt{11}$

$=( 4 -1 )\sqrt{11}$

$=3\sqrt{11}$

Simplify:

$\sqrt{66}$

The expression is already simplified.

$3 \sqrt{11}$

$2 \sqrt{11}$

$11 \sqrt{6}$

$22\sqrt{3}$

Explanation

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

$66 = 2 \times 3 \times 11$

$\sqrt{66} =\sqrt{ 2 \times 3 \times 11}$

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

Simplify the difference:

$\sqrt{70} - \sqrt{30}$

The expression cannot be simplified further.

$4\sqrt{10}$

$2\sqrt{10}$

$8\sqrt{10}$

$\sqrt{10}$

Explanation

To simplify a radical expression, first, find the prime factorization of the radicand. First, we will attempt simplify $\sqrt{70}$ as follows:

$70 = 2 \times 5 \times 7$

Since no prime factor appears twice, the expression cannot be simplified further. The same holds for $\sqrt{30}$ , since $30 = 2 \times 3 \times 5$ .

It follows that the expression $\sqrt{70} - \sqrt{30}$ is already in simplest form.

Simplify the sum:

$\sqrt{11} + \sqrt{22}+ \sqrt{33}$

The expression cannot be simplified further.

$6\sqrt{11}$

$\sqrt{66}$

$5\sqrt{11}$

$\sqrt{55}$

Explanation

To simplify a radical expression, first find the prime factorization of the radicand. First, we will attempt simplify $\sqrt{33}$ as follows:

$33 = 3 \times 11$

Since no prime factor appears twice, the expression cannot be simplified further. The same holds for $\sqrt{22}$ , since $22 = 2 \times 11$ ; the same also holds for $\sqrt{11}$ , since 11 is prime.