Exponents and the Distributive Property - ACT Math

Card 0 of 135

Question

$(4x-9)^{3}=?$

Answer

FOIL the first two terms:

Next, multiply this expression by the last term:

Finally, combine the terms:

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Question

If , what is the value of the equation $q(q-7)^{2}$ ?

Answer

Plug in for in the equation $q(q-7)^{2}$

That gives: $2(2-7)^{2}$

Then solve the computation inside the parenthesis: $2(-5)^{2}$

The answer should then be

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Question

For all $x,\ (5x+2)^{2}=$ ?

Answer

$(5x+2)^{2}$ is equivalent to $(5x+2)(5x+2)$ .

Using the FOIL method, you multiply the first number of each set $5x\cdot 5x=25x^{2}$ , multiply the outer numbers of each set $5x\cdot 2=10x$ , multiply the inner numbers of each set $2\cdot 5x=10x$ , and multiply outer numbers of each set $2\cdot 2=4$ .

Adding all these numbers together, you get $25x^{2}+10x+10x+4=25x^{2}+20x+4$ .

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Question

Square the binomial.

$(x^{2}y^{4}+xy^{6})^{2}$

Answer

$(x^{2}y^{4}+xy^{6})^{2}$

$(x^{2}y^{4}+xy^{6})(x^{2}y^{4}+xy^{6})$

We will need to FOIL.

First:

Inside: $xy^6x^2y^4=x^3y^{10}$

Outside: $x^2y^4xy^6=x^3y^{10}$

Last: $xy^6*xy^6=x^2y^{12}$

Sum all of the terms and simplify.

$x^4y^8+x^3y^{10}+x^3y^{10}+x^2y^{12}$

$x^4y^8+2x^3y^{10}+x^2y^{12}$

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Question

The expression is equivalent to __________.

Answer

Use FOIL and be mindful of exponent rules. Remember that when you multiply two terms with the same bases but different exponents, you will need to add the exponents together.

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Question

The expression is equivalent to __________.

Answer

Remember to add exponents when two terms with like bases are being multiplied.

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Question

Use the FOIL method to simplify the following expression:

Answer

Use the FOIL method to simplify the following expression:

Step 1: Expand the expression.

Step 2: FOIL

First: $x^3\cdot x^3 = x^6$

Outside: $x^3 \cdot 2x^2 =2x^5$

Inside: $2x^2 \cdot x^3 = 2x^5$

Last: $2x^2 \cdot 2x^2 = 4x^4$

Step 2: Sum the products.

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Question

The concept of FOIL can be applied to both an exponential expression and to an exponential modifier on an existing expression.

For all , = __________.

Answer

Using FOIL, we see that $(3x^3 + 7x^2)^2 = (3x^3 + 7x^2) \cdot(3x^3 + 7x^2)$

First = $3x^3 \cdot 3x^3 = 9x^6$

Outer = $3x^3 \cdot 7x^2 = 21x^5$

Inner = $7x^2 \cdot 3x^3 = 21x^5$

Last = $7x^2 \cdot 7x^2 = 49x^4$

Remember that terms with like exponents are additive, so we can combine our middle terms:

Now order the expression from the highest exponent down:

Thus,

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Question

The rule for adding exponents is $a^m + a^n = a^{m+n}$ .

The rule for multiplying exponents is $(a^m)^n = a^{m\cdot n}$ .

Terms with matching variables AND exponents are additive.

Multiply: $(a^3 + b^2c^2) \cdot (a^7 + bc^3)$

Answer

Using FOIL on $(a^3 + b^2c^2) \cdot (a^7 + bc^3)$ , we see that:

First: $a^3 \cdot a^7 = a^{10}$

Outer: $a^3 \cdot bc^3 = a^3bc^3$

Inner: $b^2c^2 \cdot a^7 = a^7b^2c^2$

Last: $b^2c^2 \cdot bc^3 = b^3c^5$

Note that the middle terms are not additive: while they share common variables, they do not share matching exponents.

Thus, we have $a^{10} + a^7b^2c^2 + a^3bc^3 + b^3c^5$ . The arrangement goes by highest leading exponent, and alphabetically in the case of the last two terms.

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Question

Simplify:

Answer

First, merely FOIL out your values. Thus:

becomes

Now, just remember that when you multiply similar bases, you add the exponents. Thus, simplify to:

Since nothing can be combined, this is the final answer.

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Question

Distribute and simplify:

Answer

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

Multiply the First terms:

$x^2\cdot x = x^3$

Multiply the Outer terms:

$x^2\cdot 8 = 8x^2$

Multiply the Inner terms:

$-1\cdot x = -x$

Multiply the Last terms:

$-1\cdot 8 = -8$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms). In this case, no two terms are compatible.

Arrange your answer in descending exponential form, and you're done.

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Question

Distribute and simplify:

Answer

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

Multiply the First terms:

$5x^2\cdot 2x = 10x^3$

Multiply the Outer terms:

$5x^2\cdot 1 = 5x^2$

Multiply the Inner terms:

$-x\cdot 2x = -2x^2$

Multiply the Last terms:

$-x\cdot 1 = -x$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms).

Arrange your answer in descending exponential form, and you're done.

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Question

Distribute and simplify:

Answer

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

Multiply the First terms:

$x^2\cdot x^2 = x^4$

Multiply the Outer terms:

$x^2\cdot x = x^3$

Multiply the Inner terms:

$x\cdot x^2 = x^3$

Multiply the Last terms:

$x\cdot x = x^2$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms).

Arrange your answer in descending exponential form, and you're done.

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Question

Distribute and simplify:

Answer

A clever eye might spot that this binomial takes the form , which means you can jump straight to as the answer. But let's look at it in detail below.

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

Multiply the First terms:

$4x^5\cdot 4x^5 = 16x^{10}$

Multiply the Outer terms:

$4x^5\cdot 4x = 16x^6$

Multiply the Inner terms:

$-4x\cdot 4x^5 = -16x^6$

Multiply the Last terms:

$-4x\cdot 4x = -16x^2$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms).

Arrange your answer in descending exponential form, and you're done.

$(4x^5-4x)(4x^5+4x) = 16x^{10} - 16x^2$

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Question

Distribute and simplify:

Answer

To FOIL this binomial distribution, we simply distribute the terms in a specific order:

Multiply the First terms:

$2x^4\cdot 6x = 12x^5$

Multiply the Outer terms:

$2x^4\cdot 3x^7 = 6x^{11}$

Multiply the Inner terms:

$-2x^3\cdot 6x = -12x^4$

Multiply the Last terms:

$-2x^3\cdot 3x^7 = -6x^{10}$

Lastly, combine any terms that allow this (usually, but not always, the two middle terms). In this case, no two terms are compatible.

Arrange your answer in descending exponential form, and you're done.

$(2x^4-2x^3)(6x+3x^7) = 6x^{11} - 6x^{10} +12x^5 - 12x^4$

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Question

$(4x-9)^{3}=?$

Answer

FOIL the first two terms:

Next, multiply this expression by the last term:

Finally, combine the terms:

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Question

If , what is the value of the equation $q(q-7)^{2}$ ?

Answer

Plug in for in the equation $q(q-7)^{2}$

That gives: $2(2-7)^{2}$

Then solve the computation inside the parenthesis: $2(-5)^{2}$

The answer should then be

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Question

For all $x,\ (5x+2)^{2}=$ ?

Answer

$(5x+2)^{2}$ is equivalent to $(5x+2)(5x+2)$ .

Using the FOIL method, you multiply the first number of each set $5x\cdot 5x=25x^{2}$ , multiply the outer numbers of each set $5x\cdot 2=10x$ , multiply the inner numbers of each set $2\cdot 5x=10x$ , and multiply outer numbers of each set $2\cdot 2=4$ .

Adding all these numbers together, you get $25x^{2}+10x+10x+4=25x^{2}+20x+4$ .

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Question

Square the binomial.

$(x^{2}y^{4}+xy^{6})^{2}$

Answer

$(x^{2}y^{4}+xy^{6})^{2}$

$(x^{2}y^{4}+xy^{6})(x^{2}y^{4}+xy^{6})$

We will need to FOIL.

First:

Inside: $xy^6x^2y^4=x^3y^{10}$

Outside: $x^2y^4xy^6=x^3y^{10}$

Last: $xy^6*xy^6=x^2y^{12}$

Sum all of the terms and simplify.

$x^4y^8+x^3y^{10}+x^3y^{10}+x^2y^{12}$

$x^4y^8+2x^3y^{10}+x^2y^{12}$

Compare your answer with the correct one above

Question

The expression is equivalent to __________.

Answer

Use FOIL and be mindful of exponent rules. Remember that when you multiply two terms with the same bases but different exponents, you will need to add the exponents together.

Compare your answer with the correct one above

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