AskSia

Plus

1.(鞍山中考)要使式子

\sqrt{2-x}

有意义, 则

\mathrm{x}

的取值范围是

(\quad)

x>0

B. $x ...

Apr 7, 2024

1.(鞍山中考)要使式子

\sqrt{2-x}

有意义, 则

\mathrm{x}

的取值范围是

(\quad)

A. x>0 B.

x \geqslant-2

x \geqslant 2

x \leqslant 2

2.矩形具有而菱形不具有的性质是( ) A.两组对边分别平行 B.对角线相等 C.对角线互相平分 D.两组对角分别相等 3.下列计算正确的是( ) A.

\sqrt{4} \times \sqrt{6}=4 \sqrt{6}

\sqrt{4}+\sqrt{6}=\sqrt{10}

\sqrt{40} \div \sqrt{5}=2 \sqrt{2}

\sqrt{(-15)^{2}}=-15

Solution by Steps

step 2

The expression inside the square root is

2-x

. For

\sqrt{2-x}

to be defined,

2-x

must be greater than or equal to 0

step 3

Solving the inequality

2-x \geq 0

gives us

x \leq 2

Key Concept

Square Root Domain

Explanation

The domain of a square root function

\sqrt{a-x}

includes all

x

values such that

a-x \geq 0

, which ensures the expression under the square root is non-negative.

Solution by Steps

step 2

A rectangle is defined by having four right angles and opposite sides that are equal and parallel

step 3

A rhombus is defined by having four sides of equal length and opposite angles that are equal, but it does not necessarily have right angles

step 4

The property that rectangles have and rhombuses do not is that the diagonals of a rectangle are equal in length, while the diagonals of a rhombus are not necessarily equal

Key Concept

Diagonals of Rectangles and Rhombuses

Explanation

In a rectangle, the diagonals are equal in length, which is not a property shared by rhombuses. Rhombuses have diagonals that bisect each other at right angles but are not necessarily equal in length.

Solution by Steps

step 2

The square root of a product

\sqrt{ab}

is equal to the product of the square roots

\sqrt{a} \times \sqrt{b}

step 3

Simplifying

\sqrt{4} \times \sqrt{6}

gives

2 \times \sqrt{6}

, not

4 \sqrt{6}

step 4

The sum of square roots

\sqrt{a} + \sqrt{b}

cannot be simplified to

\sqrt{a+b}

step 5

Simplifying

\sqrt{40} \div \sqrt{5}

gives

\sqrt{8}

, which simplifies further to

2\sqrt{2}

step 6

The square root of a square

\sqrt{a^2}

is the absolute value of

a

, so

\sqrt{(-15)^2}

15

, not

-15

Key Concept

Square Root Simplification

Explanation

The square root of a product can be expressed as the product of the square roots, and the square root of a quotient can be expressed as the quotient of the square roots. The square root of a squared number is the absolute value of the original number.