How to find the square root of a decimal - 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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