Properties and Identities - SAT Subject Test in Math II

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Question

What property of arithmetic is demonstrated below?

Answer

The statement shows that two numbers can be added in either order to achieve the same result. This is the commutative property of addition.

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Question

What property of arithmetic is demonstrated below?

Answer

The fact that 0 can be added to any number to yield the latter number as the sum is the identity property of addition.

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Question

What property of arithmetic is demonstrated below?

Answer

For every real number, there is a number that can be added to it to yield the sum 0. This is the inverse property of addition.

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Question

What property of arithmetic is demonstrated below?

$m \angle ABC = m \angle ABC$

Answer

That any number is equal to itself is the reflexive property of equality.

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Question

What property of arithmetic is demonstrated below?

If $m \angle 1 = m \angle 2$ then $m \angle 2 = m \angle 1$ .

Answer

If an equality is true, then it can be correctly stated with the expressions in either order with equal validity. This is the symmetric property of equality.

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Question

What property of arithmetic is demonstrated below?

$m \angle ABC + 45^{\circ } = 45^{\circ } + m \angle ABC$

Answer

The statement shows that two numbers can be added in either order to yield the same sum. This is the commutative property of addition.

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Question

Which expression is not equal to 0 for all positive values of ?

Answer

$\left (x+1 \right )^{0}$ is the correct choice.

for all values of , since, by the zero property of multiplication, any number multiplied by 0 yields product 0.

for all values of - this is a direct statement of the inverse property of addition.

$0^{x} = 0$ , since 0 raised to any positive power yields a result of 0.

$\left (x+1 \right )^{0} = 1$ , since any nonxero number raised to the power of 0 yields a result of 1.

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Question

Which of the following sets is not closed under multiplication?

Answer

A set is closed under multiplication if and only the product of any two (not necessarily distinct) elements of that set is itself an element of that set.

$\left { 2, 4, 6, 8, 10...\right }$ is closed under multiplication since the product of two even numbers is even:

$2N \cdot 2M = 4MN = 2 (2MN)$

$\left { 1, 4, 9, 16, 25,...\right }$ is closed under multiplication since the product of two perfect squares is a perfect square:

$N^{2} \cdot M^{2} = (NM)^{2}$

$\left { \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} ... \right }$ is closed under multiplication since the product of two square roots of positive integers is the square root of a positve integer:

$\sqrt{N} \cdot \sqrt{M} = \sqrt{NM}$

But $\left { i, 2i, 3i, 4i, 5i...\right }$ , as can be seen here, is not closed under multiplication:

$2i \cdot 3i = 2\cdot 3 \cdot i \cdot i = 6 \cdot i^{2} = 6 (-1)= -6 otin \left { i, 2i, 3i, 4i, 5i...\right }$

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