Basic Squaring / Square Roots - SAT 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

Evaluate: $\sqrt{1.44}$

Answer

The square root of a number returns a positive and negative number that multiplies by itself to obtain the number inside the square root.

$\sqrt{1.44}= \pm1.2$

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Question

Without using a calculator, solve for x:

Answer

Simplifying the given equation gives us

$x=\pm\sqrt{\frac{16}{100}}=\pm\frac{\sqrt{16}}{\sqrt{100}}$

$=\pm\frac{4}{10}=\pm0.4$

Therefore, the correct answer is $\pm0.4$ .

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Question

If and , what is the value of ?

Answer

We are given the equation . To determine the value of , take the square root of both sides.

$x=\sqrt{0.000004}$ .

To calculate this value, take note of the following pattern:

$\sqrt{4}=\pm2$

$\sqrt{0.04}=\pm0.2$

$\sqrt{0.0004}=\pm0.02$

Each succeeding radicand is $\frac{1}{100}$ of the previous radicand, and the value of each succeeding square root is $\frac{1}{10}$ of the previous value. Continuing the pattern:

$\sqrt{0.000004}=\pm0.002$ . However, we are also given the condition that ; hence, we eliminate the extraneous solution and conclude that the only valid solution of is .

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Question

Evaluate:

0.082

Answer

0.08 * 0.08

First square 8:

8 * 8 = 64

Then move the decimal four places to the left:

0.0064

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Question

If all real values of $x$ lie between 0 and 1, which of the following is always greater than 1?

Answer

If $x$ is greater than 0, then adding 1 to $x$ will make it greater than 1. Taking a number between 0 and 1 to a power results in a smaller number.

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