ACT Math : Algebra

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #14 : How To Divide Exponents

What is the simplified form for the following expression?

\(\displaystyle \frac{(2xz^2)^3}{(4zx^2)^2}\)

Possible Answers:

\(\displaystyle 2*z^3*x^3\)

\(\displaystyle \frac{2x^4}{z}\)

\(\displaystyle \frac{2x}{z^4}\)

\(\displaystyle \frac{z^4}{2x}\)

\(\displaystyle \frac{2z}{x^4}\)

Correct answer:

\(\displaystyle \frac{z^4}{2x}\)

Explanation:

Break up by variable.

\(\displaystyle \frac{2^3}{4^2}= \frac{1}{2}\)

\(\displaystyle \frac{x^3}{x^4}= \frac{1}{x}\)

\(\displaystyle \frac{z^6}{z^2} = z^4\)

Therefore the simplified form becomes,

\(\displaystyle \frac{(2xz^2)^3}{(4zx^2)^2}=\frac{z^4}{2x}\).

Example Question #15 : How To Divide Exponents

\(\displaystyle 64^{\frac{4}{3}}\) can be written as which of the following?

A. \(\displaystyle \sqrt[3]{64^4}\)

B. \(\displaystyle (\sqrt[3]{64})^{4}\)

C. \(\displaystyle 256\)

Possible Answers:

A, B and C

C only

B and C

A and C

B only

Correct answer:

A, B and C

Explanation:

C is computing the exponent, while A and B are equivalent due to properties of fractional exponents.

Remember that...

\(\displaystyle a^{\frac{b}{c}} = \sqrt[c]{a^b} = (\sqrt[c]{a})^b\)

\(\displaystyle 64^{\frac{4}{3}}=\) \(\displaystyle \sqrt[3]{64^4}=\) \(\displaystyle (\sqrt[3]{64})^{4}=\) \(\displaystyle 256\)

Example Question #1546 : Gre Quantitative Reasoning

\(\displaystyle \dpi{100} \small \frac{7^{10}-7^{8}}{7^{9}-7^{7}}=\)

Possible Answers:

\dpi{100} \small 343\(\displaystyle \dpi{100} \small 343\)

\dpi{100} \small 7\(\displaystyle \dpi{100} \small 7\)

\dpi{100} \small 28\(\displaystyle \dpi{100} \small 28\)

\dpi{100} \small 49\(\displaystyle \dpi{100} \small 49\)

\dpi{100} \small 42\(\displaystyle \dpi{100} \small 42\)

Correct answer:

\dpi{100} \small 7\(\displaystyle \dpi{100} \small 7\)

Explanation:

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is \(\displaystyle 7^7\)

\(\displaystyle \frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}\)

Now, we can cancel out the \(\displaystyle 7^7\) from the numerator and denominator and continue simplifying the expression.

\(\displaystyle \frac{7^7\left ( 7^3-7^1\right)}{7^7\left ( 7^2-1\right)}=\frac{7^3-7^1}{7^2-1}=\frac{343-7}{49-1}=\frac{336}{48}=7\)

Example Question #771 : Algebra

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

Possible Answers:

25x^{2}+10x+4\(\displaystyle 25x^{2}+10x+4\)

25x^{2}+20x+4\(\displaystyle 25x^{2}+20x+4\)

10x^{2}+4\(\displaystyle 10x^{2}+4\)

25x^{2}+4\(\displaystyle 25x^{2}+4\)

10x+4\(\displaystyle 10x+4\)

Correct answer:

25x^{2}+20x+4\(\displaystyle 25x^{2}+20x+4\)

Explanation:

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

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

Adding all these numbers together, you get 25x^{2}+10x+10x+4=25x^{2}+20x+4\(\displaystyle 25x^{2}+10x+10x+4=25x^{2}+20x+4\)

Example Question #772 : Algebra

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

 

Possible Answers:

\(\displaystyle 64x^3-432x^2+972x-729\)

\(\displaystyle 12x^3-27\)

\(\displaystyle 12x-27\)

\(\displaystyle 64x^3-729\)

Correct answer:

\(\displaystyle 64x^3-432x^2+972x-729\)

Explanation:

FOIL the first two terms:

\(\displaystyle (4x-9)(4x-9)=16x^2-36x-36x+81 = 16x^2-72x+81\)

Next, multiply this expression by the last term:

\(\displaystyle (16x^2-72x+81)(4x-9)=64x^3-144x^2-22x^2+648x+324x-729\)

Finally, combine the terms:

\(\displaystyle (4x-9)^3=64x^3-432x^2+972x-729\)

 

 

Example Question #773 : Algebra

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

 

Possible Answers:

\(\displaystyle 25\)

\(\displaystyle -50\)

\(\displaystyle 100\)

\(\displaystyle 50\)

\(\displaystyle 10\)

Correct answer:

\(\displaystyle 50\)

Explanation:

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

That gives: \(\displaystyle 2(2-7)^{2}\)

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

The answer should then be \(\displaystyle 50\)

 

 

Example Question #75 : Exponents

The expression \(\displaystyle (4x^3-2)(9x^6+7)\) is equivalent to __________.

Possible Answers:

\(\displaystyle 36x^{18}-18x^6+28x^3-14\)

\(\displaystyle 36x^9-18x^6-28x^3+14\)

\(\displaystyle 36x^9-18x^6+28x^3-14\)

\(\displaystyle 4x^9-18x^6+28x^3-14\)

\(\displaystyle 9x^9-2x^6+7x^3-14\)

Correct answer:

\(\displaystyle 36x^9-18x^6+28x^3-14\)

Explanation:

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.

Foilexpo

 

Example Question #774 : Algebra

The expression \(\displaystyle (3x^2+x)(4x^3+2x)\) is equivalent to __________.

Possible Answers:

\(\displaystyle 12x^5+4x^4+6x^3+6x^2\)

\(\displaystyle 12x^5+4x^4+8x^2\)

\(\displaystyle 12x^6+4x^3+6x^2+2x\)

\(\displaystyle 12x^5+10x^4+6x^3+2x^2\)

\(\displaystyle 12x^5+4x^4+6x^3+2x^2\)

Correct answer:

\(\displaystyle 12x^5+4x^4+6x^3+2x^2\)

Explanation:

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

 

Foilexpo

Example Question #3 : How To Use Foil With Exponents

Use the FOIL method to simplify the following expression:

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

Possible Answers:

\(\displaystyle 5x^6+4x^4\)

\(\displaystyle x^6+4x^5+4x^4\)

\(\displaystyle x^6+4x^4\)

\(\displaystyle x^6+2x^4\)

\(\displaystyle 4x^5+4\)

Correct answer:

\(\displaystyle x^6+4x^5+4x^4\)

Explanation:

Use the FOIL method to simplify the following expression:

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

Step 1: Expand the expression.

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

Step 2: FOIL

First: \(\displaystyle x^3\cdot x^3 = x^6\)

Outside: \(\displaystyle x^3 \cdot 2x^2 =2x^5\)

Inside: \(\displaystyle 2x^2 \cdot x^3 = 2x^5\)

Last: \(\displaystyle 2x^2 \cdot 2x^2 = 4x^4\)

Step 2: Sum the products.

\(\displaystyle x^6+2x^5+2x^5+4x^4\)

\(\displaystyle x^6+4x^5+4x^4\)

Example Question #1 : How To Use Foil With Exponents

The rule for adding exponents is \(\displaystyle a^m + a^n = a^{m+n}\).

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

Terms with matching variables AND exponents are additive.

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

Possible Answers:

\(\displaystyle a^{10} + a^21b^2c^6 + b^3c^5\)

\(\displaystyle a^{10} + b^3c^5\)

\(\displaystyle a^{10} + a^7b^2c^2 + a^3bc^3 + b^3c^5\)

\(\displaystyle a^{10} + a^{10}b^3c^5 + b^3c^5\)

\(\displaystyle a^{21}+b^2c^6\)

Correct answer:

\(\displaystyle a^{10} + a^7b^2c^2 + a^3bc^3 + b^3c^5\)

Explanation:

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

First: \(\displaystyle a^3 \cdot a^7 = a^{10}\)

Outer: \(\displaystyle a^3 \cdot bc^3 = a^3bc^3\)

Inner: \(\displaystyle b^2c^2 \cdot a^7 = a^7b^2c^2\)

Last: \(\displaystyle b^2c^2 \cdot bc^3 = b^3c^5\)

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

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

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