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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #251 : Radicals

Simplify $\frac{\sqrt{45}}{\sqrt{5}}$ .

Possible Answers:

$3\sqrt{5}$

$\sqrt{7}$

It cannot be simplified any further.

Correct answer:

Explanation:

The Quotient Raised to a Power rule states that $\frac{a^{x}}{b^{x}}=(\frac{a}{b})^{x}$ .

Remember that a square root is the equivalent of raising a term to the 1/2 power.

$\frac{a^{1/2}}{b^{1/2}}=\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

In this case:

$\frac{\sqrt{45}}{\sqrt{5}}=\sqrt{\frac{45}{5}}=\sqrt{9}=3$

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Example Question #1491 : Mathematical Relationships And Basic Graphs

Simplify $\frac{\sqrt{9}}{\sqrt{27}}$

Possible Answers:

$\sqrt{9}$

The expression is in simplest form

$\sqrt{3}$

$\frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{9}}$

Correct answer:

$\frac{1}{\sqrt{3}}$

Explanation:

$\frac{\sqrt{9}}{\sqrt{27}}\rightarrow\frac{\sqrt{3}\sqrt{3}}{\sqrt{3}\sqrt{3}\sqrt{3}}$

Two radicals in the numerator cancel with two of the radicals in the denominator leaving

$\frac{1}{\sqrt{3}}$

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Example Question #162 : Simplifying Radicals

$\sqrt{4x^2}\cdot \sqrt{2x^5}$

Possible Answers:

$x^3\sqrt{2x}$

$2x^2\sqrt{2x}$

$2x^3\sqrt{2}$

$2x^3\sqrt{2x}$

$2x^3\sqrt{x}$

Correct answer:

$2x^3\sqrt{2x}$

Explanation:

To multiply and simplify this expression, multiply and put everything under a big radical:

$\sqrt{8x^7}$ .

Remember that when multiplying exponents and bases are the same, add exponents. Now simplify that radical. For every pair of the same term, cross it out underneath the radical and put one outside the radical.

Therefore, your answer is:

$2x^3\sqrt{2x}.$

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Example Question #163 : Simplifying Radicals

Evaluate: $\frac{\sqrt3}{\sqrt5}\times \frac{\sqrt2}{\sqrt6}$

Possible Answers:

$\frac{6\sqrt{5}}{5}$

$\frac{\sqrt{2}}{5}$

$\frac{\sqrt{5}}{5}$

$\frac{\sqrt{5}}{3}$

$\frac{\sqrt{7}}{}{3}$

Correct answer:

$\frac{\sqrt{5}}{5}$

Explanation:

Multiply the numerator with the numerator.

$\sqrt{3} \times \sqrt{2} = \sqrt6$

Multiply the denominator with the denominator.

$\sqrt5 \times \sqrt6 = \sqrt{30}$

Divide the numerator with the denominator.

$\frac{\sqrt6}{\sqrt{30}}$

Rationalize the denominator. Multiply by square root thirty on the numerator and denominator.

$\frac{\sqrt6}{\sqrt{30}}\times \frac{\sqrt{30}}{\sqrt{30}} = \frac{\sqrt{180}}{30}$

Rewrite the numerator by common factors of a perfect square.

$\frac{\sqrt{180}}{30} = \frac{\sqrt{36}\cdot \sqrt5}{30} =\frac{6\sqrt5}{30}$

Reduce this fraction.

The answer is: $\frac{\sqrt{5}}{5}$

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Example Question #164 : Simplifying Radicals

Multiply: $\sqrt{12}\times \sqrt6 \times \sqrt3$

Possible Answers:

$6\sqrt2$

$6\sqrt6$

$3\sqrt6$

$6\sqrt3$

$12\sqrt6$

Correct answer:

$6\sqrt6$

Explanation:

It is possible to multiply all the integers together to form one radical, but doing so will give a square root of a value that will need to be factored.

Instead, rewrite each square root by their factors.

$\sqrt{12}\times \sqrt6 \times \sqrt3 =( \sqrt{4})(\sqrt{3}) \times \sqrt{6} \times \sqrt3$

A radical multiplied by itself will become the integer. Simplify the expression.

$2\times 3\times \sqrt6 = 6\sqrt6$

The answer is: $6\sqrt6$

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Example Question #1492 : Mathematical Relationships And Basic Graphs

Multiply the radicals: $\sqrt{5}\times -\sqrt{10} \times \sqrt2$

Possible Answers:

10i

$-\frac{3\sqrt5}{2}$

$-2\sqrt{5}$

$-10\sqrt2$

Correct answer:

Explanation:

Multiply the numbers in the radicals to combine as one radical.

$\sqrt{5}\times -\sqrt{10} \times \sqrt2= -\sqrt{100}$

This value can be simplified as a perfect square.

The answer is:

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Example Question #62 : Multiplying And Dividing Radicals

Multiply the radicals: $\sqrt{3}\times \sqrt{6}\times \sqrt{12}$

Possible Answers:

$2\sqrt6$

$6\sqrt6$

$4\sqrt3$

$6\sqrt3$

$12\sqrt3$

Correct answer:

$6\sqrt6$

Explanation:

In order to multiply these radicals, we are allowed to multiply all three integers to one radical, but the final term will need to be simplified.

Instead, we can pull out common factors in order to simplify the terms.

$\sqrt{6}=\sqrt{2} \cdot \sqrt3$

$\sqrt{12} = \sqrt4 \cdot \sqrt3 = 2\sqrt3$

Rewrite the expression.

$\sqrt{3}\times \sqrt{6}\times \sqrt{12}=\sqrt{3}\times(\sqrt{2} \cdot \sqrt3)\times 2\sqrt{3}$

A radical multiplied by itself will give just the integer.

$\sqrt{3}\times(\sqrt{2} \cdot \sqrt3)\times 2\sqrt{3} = 3\times \sqrt2 \times 2\sqrt3 = 6\sqrt6$

The answer is: $6\sqrt6$

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Example Question #161 : Simplifying Radicals

Divide the radicals: $\frac{2\sqrt{6}}{\sqrt{60}}$

Possible Answers:

$\frac{2\sqrt{10}}{3}$

$\frac{2\sqrt{10}}{5}$

$\frac{\sqrt{10}}{12}$

$\frac{4\sqrt3}{3}$

$\frac{\sqrt{10}}{5}$

Correct answer:

$\frac{\sqrt{10}}{5}$

Explanation:

Rationalize the denominator by multiplying the square root of 60 on the numerator and denominator.

$\frac{2\sqrt{6}}{\sqrt{60}}\cdot \frac{\sqrt{60}}{\sqrt{60}}$

Simplify the top and bottom of the fractions.

$\frac{2\sqrt{360}}{60}$

The radical can be simplified by using common factors of perfect squares.

$\sqrt{360} = \sqrt{36}\cdot \sqrt{10} = 6\sqrt{10}$

Rewrite the term.

$\frac{2\sqrt{360}}{60} =\frac{2 \cdot 6\sqrt{10}}{60} = \frac{\sqrt{10}}{5}$

The answer is: $\frac{\sqrt{10}}{5}$

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Example Question #4161 : Algebra Ii

Evaluate: $\frac{\sqrt{3}}{\sqrt7}\cdot \frac{\sqrt{14}}{\sqrt{18}}$

Possible Answers:

$\frac{\sqrt3}{3}$

$\sqrt6$

$\sqrt3$

$\frac{1}{3}$

Correct answer:

$\frac{\sqrt3}{3}$

Explanation:

Since all terms are in radicals, we can simplify the terms by using common factors.

$\frac{\sqrt{3}}{\sqrt7}\cdot \frac{\sqrt{14}}{\sqrt{18}} = \frac{\sqrt{3}}{\sqrt7}\cdot \frac{\sqrt{2}\cdot \sqrt{7}}{\sqrt{3}\cdot \sqrt{6}} =\frac{\sqrt{2}}{\sqrt{6}}=\frac{1}{\sqrt3}$

Rationalize the denominator.

$\frac{1}{\sqrt3}\cdot \frac{\sqrt3}{\sqrt3} =\frac{\sqrt3}{3}$

The answer is: $\frac{\sqrt3}{3}$

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Example Question #4162 : Algebra Ii

Multiply: $\sqrt{50}\cdot \sqrt{18}\cdot \sqrt8$

Possible Answers:

$30\sqrt3$

$30\sqrt2$

$60\sqrt{2}$

$45\sqrt{3}$

$15\sqrt{2}$

Correct answer:

$60\sqrt{2}$

Explanation:

We can simplify the radicals before expanding by multiplication.

$\sqrt{50}\cdot \sqrt{18}\cdot \sqrt8 = [\sqrt{25}\cdot \sqrt2]\cdot [\sqrt{9}\cdot \sqrt{2}]\cdot [\sqrt{4}\cdot \sqrt2]$

Simplify the radicals.

$5\sqrt2 \cdot 3\sqrt2 \cdot 2\sqrt2$

Multiply the integers together. When a square root of a number is multiplied by itself, the radical will be eliminated, giving only the integer.

$30\cdot 2 \cdot \sqrt{2}$

The answer is: $60\sqrt{2}$

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #251 : Radicals

Example Question #1491 : Mathematical Relationships And Basic Graphs

Example Question #162 : Simplifying Radicals

Example Question #163 : Simplifying Radicals

Example Question #164 : Simplifying Radicals

Example Question #1492 : Mathematical Relationships And Basic Graphs

Example Question #62 : Multiplying And Dividing Radicals

Example Question #161 : Simplifying Radicals

Example Question #4161 : Algebra Ii

Example Question #4162 : Algebra Ii

All Algebra II Resources

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