Basic Squaring / Square Roots - ACT Math

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Question

Find the square root of the following decimal:

$\sqrt{.00081}=$

Answer

The easiest way to find the square root of a fraction is to convert it into scientific notation.

$\dpi{100} \small .00081 = 8.1 \times 10^{-4}$

The key is that the exponent in scientific notation has to be even for a square root because the square root of an exponent is diving it by two. The square root of 9 is 3, so the square root of 8.1 is a little bit less than 3, around 2.8

$\dpi{100} \small \sqrt{8.1 \times 10^{-4}} \approx 2.8 \times 10^{-2} \approx 0.028$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.0049}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small 0.0049 = 49 \cdot10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root.

$\small \sqrt{0.0049} = \sqrt{49}\cdot\sqrt{10^{-4}} = 7\cdot10^{-2} = 0.07$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.025}$

Answer

This problem can be solve more easily by rewriting the decimal into scientific notation.

$\small 0.025 = 2.5 \times 10^{-2}$

Because $\small 10^{-2}$ has an even exponent, we can take the square root of it by dividing it by 2. The square root of 4 is 2, and the square root of 1 is 1, so the square root of 2.5 is less than 2 and greater than 1.

$\small \sqrt{0.025} = \sqrt{2.5}\times \sqrt{10^{-2}} = 1.58\times 10^{-1} = 0.158$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00036}$

Answer

This problem becomes much simpler if we rewrite the decimal in scientific notation

$\small 0.00036 = 3.6\times 10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can take its square root by dividing it by two. The square root of 4 is 2, and because 3.6 is a little smaller than 4, its square root is a little smaller than 2, around 1.9

$\small \sqrt{0.025} = \sqrt{3.6}\times \sqrt{10^{-4}} \approx 1.9\times 10^{-2} = 0.019$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00000025}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.00000025 = 25 \times 10^{-8}$

Because $\small \small 10^{-8}$ has an even exponent, we can divide the exponenet by 2 to get its square root. $\small 25$ is a perfect square, whose square root is $\small 5$ .

$\small \small \sqrt{0.00000025} = \sqrt{25}\times \sqrt{10^{-8}} = 5\times10^{-4} = 0.0005$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.0169}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.0169 = 169\times10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root. $\small 169$ is a perfect square, whose square root is $\small 13$ .

$\small \small \sqrt{0.0169} = \sqrt{169}\times \sqrt{10^{-4}} =13\times10^{-2} = 0.13$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00064}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.00064 = 6.4 \times10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root. The square root of 9 is 3, and the square root of 4 is two, so the square root of 6.4 is between 3 and 2, around 2.53

$\small \small \sqrt{0.00064} = \sqrt{6.4}\times\sqrt{10^{-4}} \approx 2.53 \times10^{-2} = 0.0253$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.00001}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small \small 0.00001 = 10 \times10^{-6}$

Because $\small \small 10^{-6}$ has an even exponent, we can divide the exponenet by 2 to get its square root. The square root of 9 is 3, so the square root of 10 should be a little larger than 3, around 3.16

$\small \small \sqrt{0.00001} = \sqrt{10}\times \sqrt{10^{-6}} = 3.16\times10^{-3} = 0.00316$

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Question

Find the square root of the following decimal:

$\small \sqrt{0.004}$

Answer

To find the square root of this decimal we convert it into scientific notation.

$\small \small 0.004 = 40 \times10^{-4}$

Because $\small 10^{-4}$ has an even exponent, we can divide the exponenet by 2 to get its square root. The square root of 36 is 6, so the square root of 40 should be a little more than 6, around 6.32.

$\small \small \small \sqrt{0.004} = \sqrt{40}\times\sqrt{10^{-4}} \approx 6.32 \times10^{-2} = 0.0632$

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Question

Solve for :

$\frac{x}{\sqrt{0.04}}=\sqrt{0.16}$

Answer

Just like any other equation, isolate your variable. Start by multiplying both sides by $\sqrt{0.04}$ :

$x=\sqrt{0.16}\sqrt{0.04}$

Now, this is the same as:

$x=\sqrt{0.0064}$

You know that $\sqrt{64}$ is . You can intelligently rewrite this problem as:

$x=\sqrt{0.0064}=\sqrt{\frac{1}{10\textup{,}000}64}$ , which is the same as:

$\sqrt{\frac{1}{10\textup{,}000}64}=\sqrt{\frac{1}{10\textup{,}000}}\sqrt{64}=\frac{1}{100}*8=0.08$

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Question

Find the square root of .

Answer

Rewrite the expression in radical form.

$\sqrt{0.16}$

Rewrite the decimal with factors and simplify.

$\sqrt{0.16} = \sqrt{0.4\cdot 0.4} = \sqrt{0.4}\cdot \sqrt{0.4}=0.4$

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Question

Find the square root of .

Answer

Rewrite the question in radical form.

$\sqrt{0.04}$

Split up into its common factor.

$\sqrt{0.2 \cdot 0.2} = \sqrt{0.2} \cdot \sqrt{0.2} = 0.2$

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Question

Evaluate: $-\sqrt{0.0001}$

Answer

The answer exists because the number inside the radical is not negative.

First evaluate $\sqrt{0.0001}$ by splitting the inside number by its common factor.

$\sqrt{0.0001}= \sqrt{0.01 \cdot 0.01} = 0.01$

The negative sign before the radical means that the negative is distributed after evaluating the radical.

$-(\sqrt{0.0001})= -(0.01)= -0.01$

Therefore, the answer is .

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Question

Evaluate: $\sqrt{-0.64}$

Answer

The answer does not exist since it's not possible to take the square root of negative numbers.

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Question

If $x=\sqrt{\frac{1}{0.25}}$ , which of the following is true?

Answer

$\sqrt{\frac{1}{0.25}}=\frac{\sqrt{1}}{\sqrt{0.25}}=\frac{1}{0.5}=2$

To convince yourself that is indeed the square root of , let's square .

$0.5\times0.5=0.25$ , so this is true. Since we now know that , the answer that is true given this information is .

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Question

What is $\small 0.1^{3}$ ?

Answer

You can solve this on your calculator, or think about it as a fraction $\left( \small \frac{1}{10}\right)$ to make the problem easier to do by hand. $\small 0.1^{3}=0.1\times 0.1\times 0.1=0.001$ .

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Question

Find the square of .

Answer

Squaring a decimal is identical to multiplying decimals.

Drop the decimal, multiply the number by its self, and then add up the decimal points in the original problem and put it back into your answer.

$0.12^2=0.12\cdot0.12$ (note: 4 decimal places)

$12\cdot12=144\rightarrow 0.0144$

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Question

If and , what is equal to?

Answer

First, we must figure out what is equal to.

We do .

Now to find out what is equal to, we look at $0.12\cdot 0.12$ .

We move our decimal points over so that we are dealing with only whole numbers. This gives us $12\cdot 12=144$ .

Finally, we count how many spaces we moved our decimals in total, and we move the decimal in our answer back that many spaces. To get from to we moved our decimal two spaces to the right. Because we did this for each of the values, our total spaces we moved the decimal was to the right.

Therefore, we must take and move the decimal places to the left. This gives us .

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Question

Evaluate the following:

Answer

To solve, simply multiply 0.02 by 0.02 as though the numbers are without decimals.

Then, sum the number of spaces to the right of the decimal points in your problem and include that many in your answer.

Thus, your answer is .0004.

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