Properties of operations with real numbers

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HiSET › Properties of operations with real numbers

Questions 1 - 10

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:

$\sqrt{40}$

$4\sqrt{5}$

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 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

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

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:

$\sqrt{32}$

$4\sqrt{2}$

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:

$\sqrt{120}$

$2\sqrt{30}$

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 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.

It follows that the expression $\sqrt{11} + \sqrt{22}+ \sqrt{33}$ 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.

It follows that the expression $\sqrt{11} + \sqrt{22}+ \sqrt{33}$ is already in simplest form.

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