Pre-Algebra : Operations and Properties

Study concepts, example questions & explanations for Pre-Algebra

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Example Questions

Example Question #2 : Associative Property Of Multiplication

Which of the following displays the associative property of multiplication?

Possible Answers:

\(\displaystyle a \cdot b = ab\)

\(\displaystyle a \cdot 0 = 0\)

\(\displaystyle (a \cdot b) \cdot c = a \cdot (b \cdot c)\)

\(\displaystyle a \cdot 1 = a\)

\(\displaystyle a(b \cdot c) = ab \cdot ac\)

Correct answer:

\(\displaystyle (a \cdot b) \cdot c = a \cdot (b \cdot c)\)

Explanation:

The associative property of multiplication states that you can multiply numbers together in any order, and the answer will not change.  Therefore, 

\(\displaystyle (a \cdot b) \cdot c = a \cdot (b \cdot c)\)

displays the associative property of multiplication since the two sides are equal no matter which set of parentheses you solve first.

Example Question #391 : Operations And Properties

Simplify the expression.

\(\displaystyle (-2)(x^2+x-8)\)

Possible Answers:

\(\displaystyle -2x^2-2x-16\)

\(\displaystyle -2x^2-2x+16\)

\(\displaystyle 2x^2+2x+16\)

\(\displaystyle 2x^2+2x-16\)

Correct answer:

\(\displaystyle -2x^2-2x+16\)

Explanation:

Use the distributive property to multiply each term of the polynomial by \(\displaystyle \small -2\). Be careful to distribute the negative as well.

\(\displaystyle (-2)(x^2)+(-2)(x)-(-2)(8)\)

\(\displaystyle (-2x^2)+(-2x)-(-16)\)

\(\displaystyle -2x^2-2x+16\)

Example Question #1 : How To Use The Distributive Property In Pre Algebra

Find the value of \(\displaystyle 2(3-2) -4(2+5-7)\).

Possible Answers:

2

-6

4

6

-2

Correct answer:

2

Explanation:

We can seperate the problem into two steps:

\(\displaystyle 2(3-2) = 2(1)=2\)

\(\displaystyle -4(2+5-7) = -4(0)=0\)

We then combine the two parts:

\(\displaystyle 2+0=2\)

 

Example Question #392 : Operations And Properties

Distribute \(\displaystyle -4(5x+15y-9)\).

Possible Answers:

\(\displaystyle 20x+60y-36\)

\(\displaystyle 20x-60y+36\)

\(\displaystyle -20x-60y-36\)

\(\displaystyle -20x-60y+36\)

Correct answer:

\(\displaystyle -20x-60y+36\)

Explanation:

When distributing with negative numbers we must remember to distribute the negative to all of the variables in the parentheses.

Distribute the \(\displaystyle -4\) through the parentheses by multiplying it with each object in the parentheses to get \(\displaystyle ((-4)5x+(-4)15y-(-4)9)\).

Perform the multiplication remembering the positive/negative rules to get \(\displaystyle -20x-60y+36\), our answer.

Example Question #1 : How To Use The Distributive Property In Pre Algebra

Simplify the expression.

\(\displaystyle (-x+2)(-xy)\)

Possible Answers:

\(\displaystyle 2x^3y^2\)

\(\displaystyle x^2y-2xy\)

\(\displaystyle -x^2y-2xy\)

\(\displaystyle -2x^3y^2\)

Correct answer:

\(\displaystyle x^2y-2xy\)

Explanation:

Multiply the mononomial by each term in the binomial, using the distributive property.

\(\displaystyle (-x+2)(-xy)\)

\(\displaystyle (-xy)(-x)+(-xy)(2)\)

\(\displaystyle x^2y+(-2xy)\)

\(\displaystyle x^2y-2xy\)

 

Example Question #1 : High School Math

Simplify the expression.

\(\displaystyle 2x(5+4x+y)\)

Possible Answers:

\(\displaystyle 10x+6x^2+2xy\)

\(\displaystyle 5+6x+y\)

\(\displaystyle 10x+8x^2+2xy\)

\(\displaystyle 10+8x+2y\)

\(\displaystyle 10x+8x^2+y\)

Correct answer:

\(\displaystyle 10x+8x^2+2xy\)

Explanation:

\(\displaystyle 2x(5+4x+y)\)

Use the distributive property to multiply each term by \(\displaystyle \small 2x\).

\(\displaystyle 2x(5)+2x(4x)+2x(y)\)

Simplify.

\(\displaystyle 10x+8x^2+2xy\)

Example Question #1 : Distributive Property

Distribute:

\(\displaystyle -3(-4x+5y-8)\)

Possible Answers:

\(\displaystyle 12x+15y-24\)

\(\displaystyle 12x-15y+24\)

\(\displaystyle 12x+15y+24\)

\(\displaystyle -12x+15y-24\)

\(\displaystyle x-7y+24\)

Correct answer:

\(\displaystyle 12x-15y+24\)

Explanation:

When distributing with negative numbers we must remember to distribute the negative to all of the terms in the parentheses.

Remember, a negative multiplied by a negative is positive, and a negative multiplied by a positive number is negative.

Distribute the \(\displaystyle -3\) through the parentheses:

\(\displaystyle -4x(-3)+5y(-3)-8(-3)\)

Perform the multiplication, remembering the positive/negative rules:

\(\displaystyle 12x-15y+24\)

 

Example Question #1 : High School Math

Which of the following is equivalent to \(\displaystyle -3(2d - 4)\)?

Possible Answers:

\(\displaystyle -6d + 4\)

\(\displaystyle -6d - 12\)

\(\displaystyle 6d - 12\)

\(\displaystyle -6d - 4\)

\(\displaystyle -6d + 12\)

Correct answer:

\(\displaystyle -6d + 12\)

Explanation:

We need to distribute -3 by multiplying both terms inside the parentheses by -3.:

 \(\displaystyle -3(2d - 4) = -3(2d) + (-3)(-4)\).

Now we can multiply and simplify. Remember that multiplying two negative numbers results in a positive number:

\(\displaystyle -3(2d)+(-3)(-4)= -6d + 12\)

Example Question #1 : Distributive Property

Expand:

\(\displaystyle -4x(3x^2-7x+2)\)

Possible Answers:

\(\displaystyle 12x^3+28x-8\)

\(\displaystyle -12x^3-28x-8x\)

\(\displaystyle 12x^3-28x+8x\)

\(\displaystyle -12x^3+28x^{2}-8x\)

\(\displaystyle 12x^3+28x-8x\)

Correct answer:

\(\displaystyle -12x^3+28x^{2}-8x\)

Explanation:

\(\displaystyle \small -4x(3x^2-7x+2)\)

Use the distributive property. Do not forget that the negative sign needs to be distributed as well!

\(\displaystyle \small \small (-4x)(3x^2)=-12x^3\)

\(\displaystyle \small (-4x)(-7x)=28x^2\)

\(\displaystyle \small (-4x)(2)=-8x\)

Add the terms together:

\(\displaystyle \small -12x^3+28x^2+(-8x)=-12x^3+28x^2-8x\)

Example Question #1 : Distributive Property

Distribute:

\(\displaystyle -5(x+15)\)

Possible Answers:

\(\displaystyle -5x+75\)

\(\displaystyle 5x+75\)

\(\displaystyle -5x+15\)

\(\displaystyle -5x-75\)

\(\displaystyle -5x-15\)

Correct answer:

\(\displaystyle -5x-75\)

Explanation:

Remember that a negative multiplied by a negative is positive, and a negative multiplied by a positive is negative.

Distribute the \(\displaystyle -5\) through the parentheses by multiplying it by each of the two terms: 

\(\displaystyle -5x-75\)

 

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