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ACT Math : Arithmetic

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Example Question #11 : Exponents And Roots

Simplify: $8\sqrt{136}$

Possible Answers:

$24\sqrt{3}$

$16\sqrt{8}$

$8\sqrt{136}$

$16\sqrt{34}$

$10\sqrt{34}$

Correct answer:

$16\sqrt{34}$

Explanation:

Again here, if no perfect square is easily recognized try dividing by , , or .

$8\sqrt{136} = 8\sqrt{4 \cdot 34} = 16 \sqrt{34}$

Note that the we obtained by simplifying $\sqrt{4}$ is multiplied, not added, to the already outside the radical.

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Example Question #811 : New Sat

Simplify:

$\sqrt{48}$

Possible Answers:

$\sqrt{48}$

$4\sqrt3$

$3\sqrt{4}$

Correct answer:

$4\sqrt3$

Explanation:

To solve, simply find a perfect square factor and pull it out of the square root.

Recall the factors of 48 include (16, 3). Also recall that 16 is a perfect square since 4*4=16.

Thus,

$\sqrt{48}\Rightarrow \sqrt{16*3}\Rightarrow \sqrt{4^2*3}\Rightarrow 4\sqrt{3}$

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Example Question #17 : Simplifying Square Roots

Solve:

$\sqrt{20}+\sqrt{45}=?$

Possible Answers:

$\sqrt{65}$

$2\sqrt{5}$

$5\sqrt{20}$

$5\sqrt{5}$

$4\sqrt{5}$

Correct answer:

$5\sqrt{5}$

Explanation:

The trick to these problems is to simplify the radical by using the following rule: $\sqrt{ab}=\sqrt{a}*\sqrt{b}$ and $a\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b}$ Here, we need to find a common factor for the radical. This turns out to be five because $20=5*4\textup{ and }45=9*5.$ Remember, we want to include factors that are perfect squares, which are what nine and four are. Therefore, we can rewrite the equation as: $\sqrt{20}+\sqrt{45}=2\sqrt{5}+3\sqrt{5}=5\sqrt{5}.$

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Example Question #1 : Arithmetic

Simplify:

$\sqrt[2]{24,300}$

Possible Answers:

$10\sqrt[2]{243}$

$90\sqrt[2]{3}$

$9\sqrt[2]{300}$

$90\sqrt[2]{270}$

$900\sqrt[2]{3}$

Correct answer:

$90\sqrt[2]{3}$

Explanation:

$\sqrt[2]{24,300}$

When simplifying the square root of a number that may not have a whole number root, it's helpful to approach the problem by finding common factors of the number inside the radicand. In this case, the number is 24,300.

What are the factors of 24,300?

24,300 can be factored into:

$24,300: 243\cdot 100$

When there are factors that appear twice, they may be pulled out of the radicand. For instance, 100 is a multiple of 24,300. When 100 is further factored, it is $10^{2}$ (or 10x10). However, 100 wouldn't be pulled out of the radicand, but the square root of 100 because the square root of 24,300 is being taken. The 100 is part of the24,300. This means that the problem would be rewritten as: $10\sqrt[2]{243}$

But 243 can also be factored: $24,300: (3\cdot 81)\cdot100$
$24,300: {\color{Orange} 9}\cdot{\color{Orange} 9}\cdot {\color{Cyan} 10}\cdot {\color{cyan} 10}\cdot 3$
Following the same principle as for the 100, the problem would become
$9\cdot 10\sqrt[2]{3}$ because there is only one factor of 3 left in the radicand. If there were another, the radicand would be lost and it would be 9*10*3.
9 and 10 may be multiplied together, yielding the final simplified answer of
$90\sqrt[2]{3}$

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Example Question #1 : Arithmetic

$\sqrt{180} + \sqrt{125} = ?$

Possible Answers:

17.5

25.0

$11\sqrt{10}$

$11\sqrt{5}$

$\sqrt{305}$

Correct answer:

$11\sqrt{5}$

Explanation:

To solve the equation $\sqrt{180} + \sqrt{125} = ?$ , we can first factor the numbers under the square roots.

$\sqrt{2\cdot 2\cdot 3\cdot 3\cdot 5} + \sqrt{5\cdot 5\cdot 5} = ?$

When a factor appears twice, we can take it out of the square root.

$2\cdot 3\sqrt{5} + 5\sqrt{5} = ?$

$6\sqrt{5} + 5\sqrt{5} = ?$

Now the numbers can be added directly because the expressions under the square roots match.

$6\sqrt{5} + 5\sqrt{5} = 11\sqrt{5}$

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Example Question #1 : How To Find The Square Of An Integer

If (x - 3)² = 64, what is a possible value of x?

Possible Answers:

-8

-5

Correct answer: -5

Explanation:

In order to solve for x, we must first take the square root of both sides.

(x - 3)² = 64

(x - 3) = +/- 8

The square root of 64 is either 8 or -8. We then solve the equation with both possible values of the square root of 64.

x - 3 = 8 (Add 3 to both sides.)

x = 11

x - 3 = -8 (Add 3 to both sides.)

x = -5

x = 11 or -5. -5 is the only possible value of x that is an answer choice.

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Example Question #1 : How To Find The Square Of An Integer

Solve: 3³

Possible Answers:

Correct answer:

Explanation:

When using a cube root. 3³ becomes 3 * 3 * 3.

3 * 3 = 9

9 * 3 = 27

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Example Question #2 : How To Find The Square Of An Integer

When you square the following numbers, which one results in an irrational number?

Possible Answers:

$\frac{1}{3}$

$\frac{7}{17}$

$\frac{1}{2}$

$\sqrt2$

$\pi$

Correct answer:

$\pi$

Explanation:

A rational number is any number than can be expressed as a fraction or a quotient of two integers. The square of π is still an irrational number. (√(2))² = 2, (1/3)² = 1/9, (7/17)²= 49/289, (1/2)² = 1/4, all can be writen in fractional form, and are thus, rational numbers.

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Example Question #83 : Arithmetic

How much larger is the sum of the squares of –2, –3, and 4 than the sum of these integers?

Possible Answers:

–1

Infinitely larger

Correct answer:

Explanation:

The square of –2 is 4, of –3 is 9, and of 4 is 16.

The sum of these squares is 29.

The sum of (–2) + (–3) + (4) is –1.

29 – (–1) = 30

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Example Question #81 : Arithmetic

$\dpi{100} \small \sqrt{15}$ is between what two integers?

Possible Answers:

$\dpi{100} \small 4$ and $\dpi{100} \small 5$

$\dpi{100} \small 7$ and $\dpi{100} \small 8$

$\dpi{100} \small 5$ and $\dpi{100} \small 6$

$\dpi{100} \small 2$ and $\dpi{100} \small 3$

$\dpi{100} \small 3$ and $\dpi{100} \small 4$

Correct answer:

$\dpi{100} \small 3$ and $\dpi{100} \small 4$

Explanation:

$\dpi{100} \small 3\times 3=9$

$\dpi{100} \small 4\times 4=16$

falls between and

Therefore, $\dpi{100} \small \sqrt{15}$ is between and

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ACT Math : Arithmetic

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #11 : Exponents And Roots

Example Question #811 : New Sat

Example Question #17 : Simplifying Square Roots

Example Question #1 : Arithmetic

Example Question #1 : Arithmetic

Example Question #1 : How To Find The Square Of An Integer

Example Question #1 : How To Find The Square Of An Integer

Example Question #2 : How To Find The Square Of An Integer

Example Question #83 : Arithmetic

Example Question #81 : Arithmetic

All ACT Math Resources

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