How to find the square root of a decimal - GRE Quantitative Reasoning

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Question

Find the square root of the following decimal:

$$\sqrt{.00081}$=$

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Answer

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

.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

$\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}$$

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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}$$

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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}$$

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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}$$

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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}$$

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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}$$

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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}$$

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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}$$

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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 :

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

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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 the following decimal:

$$\sqrt{.00081}$=$

Tap to reveal answer

Answer

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

.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

$\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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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}$$

Tap to reveal answer

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 :

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

Tap to reveal answer

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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