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Pre-Algebra : Identities and Properties

Study concepts, example questions & explanations for Pre-Algebra

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Example Questions

Example Question #1 : Multiplicative Inverse Property

Simplify.

$\frac{\frac{12}{c}}{\frac{3}{c^2}}$

Possible Answers:

$\frac{1}{4c}$

$\frac{4}{c}$

$\frac{3}{4c^2}$

Correct answer:

Explanation:

To simplify a compound fraction, multiply the numerator by the reciprocal of the denominator. Remember that a compound fraction can easily be rewritten as a division problem!

$\frac{\frac{12}{c}}{\frac{3}{c^2}}=\frac{12}{c}\div\frac{3}{c^2}=\frac{12}{c}\times\frac{c^2}{3}$

Solve the multiplication.

$\frac{12}{c}\times\frac{c^2}{3}=\frac{12\times c^2}{c\times3}=\frac{12c^2}{3c}$

Now we need to reduce the fraction to find our final answer.

$\frac{12c^2}{3c}=\frac{3c(4c)}{3c}=4c$

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Example Question #2 : Multiplicative Inverse Property

What is the multiplicative inverse of $\frac{x^{2}}{x +\sqrt{3}}$ where $\left ( \frac{x^{2}}{x + \sqrt{3}} \neq 0\right )$

Possible Answers:

$\frac{x^{2} + \sqrt{3}}{x}$

$\frac{x + \sqrt{3}}{x^{2}}$

$\frac{x^{2}}{x + \sqrt{3}}$

Correct answer:

$\frac{x + \sqrt{3}}{x^{2}}$

Explanation:

The rule for Multiplicative Inverse Property is $a \times \frac{1}{a} = 1$ where $\left ( a \neq 0\right )$ .

Using this rule, if

$a = \frac{x^{2}}{x + \sqrt{3}}$ ,

then $\frac{1}{a}$ is the Mulitplicative inverse, which is $\frac{1}{\frac{x^{2}}{x +\sqrt{3}}}$ .

After you simplify you get $\frac{x + \sqrt{3}}{x^{2}}$ which is the Multiplicative Inverse.

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Example Question #3 : Multiplicative Inverse Property

What is the multiplicative inverse of $-\frac{1}{\sqrt{x^{3}}}$ where $\left ( x > 0\right )$ ?

Possible Answers:

$\frac{1}{^{\sqrt{x^{3}}}}$

$-\sqrt{x^{3}}$

$\sqrt{x^{3}}$

Correct answer:

$-\sqrt{x^{3}}$

Explanation:

The rule for Multiplicative Inverse Property is

$a \times \frac{1}{a} = 1$

where $\left ( a \neq 0\right )$ .

Using this rule, if

$a = -\frac{1}{\sqrt{x^{3}}}$ ,

then $\frac{1}{a}$ is the mulitplicative inverse, which is $\frac{1}{-\frac{1}{\sqrt{x^{3}}}}$ .

After you simplify you get $-\sqrt{x^{3}}$ which is the multiplicative inverse.

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Example Question #1 : Multiplicative Inverse Property

Which of the following statements demonstrates the inverse property of multiplication?

Possible Answers:

None of the examples in the other responses demonstrates the inverse property of multiplication.

$\frac{6}{7} \times\frac{1}{4} = \frac{1}{4} \times \frac{6}{7}$

$\frac{6}{7} \times\left ( \frac{1}{4} \times \frac{3}{8} \right ) =\left ( \frac{6}{7} \times\frac{1}{4} \right ) \times \frac{3}{8}$

$\frac{6}{7} \times \frac{7}{6} = 1$

$\frac{6}{7} \times 1 = \frac{6}{7}$

Correct answer:

$\frac{6}{7} \times \frac{7}{6} = 1$

Explanation:

The inverse property of multiplication states that for every real number, a number exists, called the multiplicative inverse, such that the number and its inverse have product 1. Of the statements given, only

$\frac{6}{7} \times \frac{7}{6} = 1$

demonstrates this property.

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Example Question #2 : Multiplicative Inverse Property

Which of the following displays the multiplicative inverse property?

Possible Answers:

$a \cdot 0 = 0$

$a \cdot -1 = -a$

$a \cdot 1 = a$

$a \cdot \frac{1}{a} = 1$

$a \cdot a = a^2$

Correct answer:

$a \cdot \frac{1}{a} = 1$

Explanation:

The mulitplicative inverse property deals with reciprocals. For example, the multiplicative inverse, or reciprocal, of the number 7 is $\frac{1}{7}$ .

The multiplicative inverse property states that a number times its multiplicative inverse equals 1.

Therefore,

$a \cdot \frac{1}{a} = 1$

displays the multiplicative inverse property.

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Example Question #1 : Other Pre Algebra Properties

Simplify.

$x^{3}+y^{2}$

Possible Answers:

$x^{6}$

$x^{3}+y^{2}$

$x^{^{5}}$

Correct answer:

$x^{3}+y^{2}$

Explanation:

Since the two powers are not of the same base, we cannot combine the exponents.

We cannot simplify the equation further.

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Example Question #2 : Other Pre Algebra Properties

Simplify.

(a+b)(a-b) +b(b+1)

Possible Answers:

a^2+b^2+b

a^2+b^2+1

a^2+2b

a^2-b^2+2b

a^2+b

Correct answer:

a^2+b

Explanation:

(a+b)(a-b) +b(b+1)

The first part is (a+b)(a-b) .

One way to simplify this portion is to apply the distributive property twice to get:

a(a-b)+b(a-b)

a^2-ab+ba-b^2

By the commutative property of multiplication ab=ba , so the two middle terms cancel out (one is positive and the other negative, resulting in zero).

This leaves a^2-b^2 .

Note: the identity (a+b)(a-b)=a^2-b^2 is worth memorizing, because it is very useful in higher math. The same result can be achieved by using FOIL.

Returning to our original expression, we can substitute:

(a+b)(a-b) +b(b+1)=(a^2-b^2)+b(b+1)

Now, apply the distributive property to the second part of the expression:

$b(b+1)=(b\times b)+(b\times1)=b^2+b$

Combining everything together:

(a^2-b^2)+(b^2+b)

Use the associative property of addition to group like terms and simplify.

a^2+(-b^2+b^2)+b

a^2+0+b

a^2+b

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Example Question #3 : Other Pre Algebra Properties

What property is applied to the following expression?

$3x\left ( a^{2} + ab + \sqrt{7}\right) = 3xa^{2} + 3xab + 3x\sqrt{7}$

Possible Answers:

Associative Property of Addition

Associative Property of Multiplication

Distributive Property

Commutative Property of Multiplication

Multiplicative Inverse Property

Correct answer:

Distributive Property

Explanation:

The rule for Distributive Property is $a\left( b + c + d\right ) = ab + ac + ad$

Expression given in the question is:

$3x\left ( a^{2} + ab + \sqrt{7}\right) = 3xa^{2} + 3xab + 3x\sqrt{7}$

Hence it is Distributive Property.

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Example Question #1 : Read, Write, And Compare Decimals To Thousandths: Ccss.Math.Content.5.Nbt.A.3

What digit is in the ones place?

497.123

Possible Answers:

Correct answer:

Explanation:

The ones place will always be the first digit to the left of the decimal point.

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Example Question #4 : Other Pre Algebra Properties

What digit is in the hundredths place?

564.391

Possible Answers:

Correct answer:

Explanation:

The hundredths place will always be the second digit to the right of the decimal point.

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Pre-Algebra : Identities and Properties

Study concepts, example questions & explanations for Pre-Algebra

All Pre-Algebra Resources

Example Questions

Example Question #1 : Multiplicative Inverse Property

Example Question #2 : Multiplicative Inverse Property

Example Question #3 : Multiplicative Inverse Property

Example Question #1 : Multiplicative Inverse Property

Example Question #2 : Multiplicative Inverse Property

Example Question #1 : Other Pre Algebra Properties

Example Question #2 : Other Pre Algebra Properties

Example Question #3 : Other Pre Algebra Properties

Example Question #1 : Read, Write, And Compare Decimals To Thousandths: Ccss.Math.Content.5.Nbt.A.3

Example Question #4 : Other Pre Algebra Properties

All Pre-Algebra Resources

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