Square Roots - Algebra II

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Question

Simplify by rationalizing the denominator:

$\frac{4}{\sqrt{11} - \sqrt{3}}$

Answer

Multiply the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{11}+ \sqrt{3}$ . Then take advantage of the distributive properties and the difference of squares pattern:

$\frac{4}{\sqrt{11} - \sqrt{3}}$

$= \frac{4\left ( \sqrt{11}+ \sqrt{3} \right )}{\left (\sqrt{11} - \sqrt{3} \right ) \left ( \sqrt{11}+ \sqrt{3} \right )}$

$= \frac{4\left ( \sqrt{11}+ \sqrt{3} \right )}{\left (\sqrt{11}\right ) ^{2} -\left ( \sqrt{3} \right ) ^{2} }$

$= \frac{4\left ( \sqrt{11}+ \sqrt{3} \right )}{11- 3 }$

$= \frac{4\left ( \sqrt{11}+ \sqrt{3} \right )}{8 }$

$= \frac{ \sqrt{11}+ \sqrt{3} }{2 }$

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Question

Estimate the square root of to the nearest tenth.

Answer

Recall that we are looking for a number that, when multiplied by itself, yields . We look for the perfect squares surrounding and we find and . Thus, we know that our number must be between and is much closer to and thus it will be very close to but still less, namely .

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Question

$\sqrt{{x}^{53}}$ is a real number for what values of ?

Answer

Taking the square root of a positive number will give you a positive number, and raising any positive number to any power will result in a positive number.

Taking the square root of a negative number will result in the square root of the absolute value of the number, times i

ex. if $x=\sqrt{-9}, = 3i$

When an expression has $i^{1}$ , raising the expression to an even power will get rid of the imaginary number "i", and make the answer negative. Negative numbers are real numbers that belong on the number line.

ex. $i^{2} = -1 , i^{4} = -1, 1^{42}=-1$

However, because the expression is raised to the 53rd power, it keeps the "i" in the expression for any negative value of x. Therefore, only positive numbers (and 0 which is not included in the answer choices) satisfy the requirements for being a real number.

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Question

Estimate the square root of 110 to the nearest tenth without using a calculator`

Answer

The squares that are closest to 110 are 100 and 121. The square root of 100 is 10, and the square root of 121 is 11. 110 is almost right in the middle, which makes the answer 10.4.

10.9 is too high, because that would make the square closer to the 119-120 range.

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Question

Simplify:

$\sqrt{256x^5y^6z^3}+\sqrt{169x^3y^2}$

Answer

To simplify, we must find the squares that are underneath each radical (this can be easier to see after some factoring):

$\sqrt{256\cdot x^4\cdot x\cdot y^6\cdot z^2\cdot z} + \sqrt{169\cdot x^2\cdot x\cdot y^2}$

The squares are easier to identify now! We can pull them out of the radical after taking their square root, leaving behind everything that is not a square:

$16x^2y^3z\sqrt{xz}+13xy\sqrt{x}$

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Question

Evaluate: $(\frac{\sqrt{64}\cdot \sqrt{36}}{\sqrt{16}})^3$

Answer

Simplify the radicals inside the parentheses.

$(\frac{\sqrt{64}\cdot \sqrt{36}}{\sqrt{16}})^3 = (\frac{8\cdot 6}{4})^3$

Simplify the terms by order of operations. Solve the terms inside the parentheses first.

$(\frac{8\cdot 6}{4})^3= (2\cdot 6)^3 = (12)^3 = 12\times 12 \times 12$

The answer is:

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Question

Simplify the radicals: $\frac{2\sqrt{24}}{3\sqrt{3}}$

Answer

Rewrite the numerator by taking a square root of three as a common denominator. This way, we do not have to multiply $\frac{\sqrt3}{\sqrt3}$ to rationalize the denominator.

$\frac{2\sqrt{24}}{3\sqrt{3}} = \frac{2\sqrt{3}\cdot \sqrt{8}}{3\sqrt{3}}$

Notice that now we can eliminate the radical from the denominator. Fully simplify and rewrite the numerator.

$2\sqrt8 = 2\cdot \sqrt4 \cdot \sqrt 2 = 2(2)\sqrt2= 4\sqrt2$

Divide this by three since there is a lone three in the denominator.

The answer is: $\frac{4\sqrt2}{3}$

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Question

Evaluate the product of square roots: $\sqrt{3}\times \sqrt{18}$

Answer

We can rewrite the expression by using common factors.

$\sqrt{3}\times \sqrt{18} = \sqrt{3}\times \sqrt{9}\times \sqrt{2}$

The radical square root nine is a perfect square. The other two radicals can be multiplied together to form one radical.

$\sqrt{3}\times \sqrt{9}\times \sqrt{2} =3\times \sqrt{6}$

The answer is: $3 \sqrt6$

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Question

Evaluate the radical: $\sqrt{8}\times \sqrt{40} \times \sqrt{24}$

Answer

In order to get the most simplified answer, do not multiply all the numbers together and combine as one radical.

Rewrite each radicals in their most simplified form.

$\sqrt8 = \sqrt4 \cdot \sqrt2 = 2\sqrt2$

$\sqrt{40} = \sqrt{4} \cdot \sqrt{10} = 2\sqrt{10}$

$\sqrt{24}= \sqrt{4}\cdot \sqrt 6 = 2\sqrt6$

Multiply the terms.

$2\sqrt{2} \cdot 2\sqrt{10} \cdot2\sqrt{6} = (2 \cdot\sqrt2)(2 \cdot\sqrt2 \cdot \sqrt5)(2 \cdot\sqrt2 \cdot \sqrt3)$

Notice that multiplying a square root of a number by itself will leave only the integer and will eliminate the radical.

$\sqrt2 \cdot \sqrt2 =2$

The terms become:

$=(2 \cdot\sqrt2)(2 \cdot\sqrt2 \cdot \sqrt5)(2 \cdot\sqrt2 \cdot \sqrt3)$

$= 2\cdot2\cdot2\cdot2 (\sqrt2 \cdot \sqrt5\cdot \sqrt3)$

$=16\sqrt{30}$

The answer is: $16\sqrt{30}$

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Question

Evaluate the square root: $-3\sqrt3 + 6\sqrt3+15\sqrt3$

Answer

The coefficients of the terms share the same square root. This means that the terms can be combined as a single square root.

Add the coefficients.

$-3\sqrt3 + 6\sqrt3+15\sqrt3 = 18\sqrt3$

The answer is: $18\sqrt3$

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Question

Simplify: $\sqrt{-72}$

Answer

Rewrite the radical using common factors.

$\sqrt{-72}=\sqrt{-1}\cdot\sqrt{9}\cdot\sqrt{4}\cdot \sqrt{2}$

Recall that $\sqrt{-1}$ is equivalent to the imaginary term .

Simplify the roots.

$i\cdot 3\cdot 2\cdot \sqrt2$

The answer is: $6i\sqrt2$

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Question

Simplify the radicals: $\sqrt{16}+\sqrt{36}- \sqrt[4]{16}$

Answer

Simplify each radical. A number inside the radical means that we are looking for a number times itself that will equal to that value inside the radical.

$\sqrt{16}=4$

$\sqrt{36}=6$

For a fourth root term, we are looking for a number that multiplies itself four times to get the number inside the radical.

$\sqrt[4]{16} =2$

Replace the values and determine the sum.

$\sqrt{16}+\sqrt{36}- \sqrt[4]{16} = 4+6-2 = 8$

The answer is:

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Question

Evaluate: $-\sqrt{121}+\sqrt{64}-\sqrt{16}$

Answer

Evaluate each square root. The square root of the number is equal to a number multiplied by itself.

$-\sqrt{121}+\sqrt{64}-\sqrt{16} =-11+8-4 =-7$

The answer is:

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Question

Simplify the radicals: $\sqrt{144\times 25\times 36}$

Answer

Do not multiply the terms inside the radical. Instead, the terms inside the radical can be simplified term by term.

$\sqrt{144\times 25\times 36} = \sqrt{144}\times \sqrt{25}\times \sqrt{36}$

Simplify each square root.

$12\times 5\times 6 = 360$

The answer is:

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Question

Solve: $2\sqrt{36}-4\sqrt{100}$

Answer

Solve by evaluating the square roots first.

$\sqrt{36} = 6$

$\sqrt{100}=10$

Substitute the terms back into the expression.

$2\sqrt{36}-4\sqrt{100} = 2(6)-4(10) = 12-40= -28$

The answer is:

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Question

Solve the square roots: $\sqrt9+\sqrt{64}+\sqrt{36+64}$

Answer

Evaluate each radical. The square root of a certain number will output a number that will equal the term inside the radical when it's squared.

$\sqrt{9}=3$

$\sqrt{64}=8$

$\sqrt{36+64} = \sqrt{100} =10$

Replace all the terms.

$\sqrt9+\sqrt{64}+\sqrt{36+64} = 3+8+10 = 21$

The answer is:

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Question

Evaluate: $\sqrt{-200}$

Answer

This expression is imaginary. To simplify, we will need to factor out the imaginary term $i=\sqrt{-1}$ as well as the perfect square.

$\sqrt{-200} = \sqrt{-1}\cdot \sqrt{100}\cdot \sqrt2$

Simplify the terms.

$i\cdot 10\cdot \sqrt2$

The answer is: $10i\sqrt2$

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Question

Solve: $\sqrt{81}+\sqrt{225}-\sqrt{10000}$

Answer

Evaluate each square root. The square root identifies a number that multiplies by itself to equal the number inside the square root.

$\sqrt{81}+\sqrt{225}-\sqrt{10000} = 9+15-100$

Determine the sum.

The answer is:

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Question

Evaluate, if possible: $2\sqrt{-64}-3\sqrt{-16}$

Answer

The negative numbers inside the radical indicates that we will have imaginary terms.

Recall that $i=\sqrt{-1}$ .

Rewrite the radicals using $\sqrt{-1}$ as the common factor.

$2\sqrt{-64}-3\sqrt{-16} = 2\cdot \sqrt{-1}\cdot \sqrt{64}-3\cdot \sqrt{-1}\cdot\sqrt{16}$

Replace the terms and evaluate the square roots.

$2\cdot i\cdot8 - 3 \cdot i \cdot4 =16i-12i = 4i$

The answer is:

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Question

Simplify: $\sqrt{100}-\sqrt{64}-\sqrt{169}$

Answer

Evaluate by solving each square root first. The square root of a number is a number that multiplies by itself to achieve the number inside the square root.

$\sqrt{100} =10$

$\sqrt{64}=8$

$\sqrt{169} = 13$

Rewrite the expression.

$\sqrt{100}-\sqrt{64}-\sqrt{169} = 10-8-13$

The answer is:

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