Algebra II - High School Math

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Question

$f(x)=x^{2}-4x+7$

Find the vertex of the parabola by completing the square.

Answer

To find the vertex of a parabola, we must put the equation into the vertex form:

$\small f(x)=a[b(x-h)]^{2}+k$

The vertex can then be found with the coordinates (h, k).

To put the parabola's equation into vertex form, you have to complete the square. Completing the square just means adding the same number to both sides of the equation -- which, remember, doesn't change the value of the equation -- in order to create a perfect square.

Start with the original equation:

$f(x)=x^{2}-4x+7$

Put all of the terms on one side:

$f(x)-7=x^{2}-4x$

Now we know that we have to add something to both sides in order to create a perfect square:

$f(x)+7+\square =x^{2}-4x+\square$

In this case, we need to add 4 on both sides so that the right-hand side of the equation factors neatly.

$f(x)-7+4=x^{2}-4x+4$

Now we factor:

$f(x)-3=(x-2)^{2}$

Once we isolate , we have the equation in vertex form:

$f(x)=(x-2)^{2}+3$

Thus, the parabola's vertex can be found at $\small (2,3)$ .

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Question

Complete the square:

Answer

Begin by dividing the equation by and subtracting from each side:

$x^2+\frac{3}{2}x = -1$

Square the value in front of the and add to each side:

$x^2+\frac{3}{2}x +\frac{9}{16} = -1+\frac{9}{16}$

Factor the left side of the equation:

$(x+\frac{3}{4})^2 = \frac{-7}{16}$

Take the square root of both sides and simplify:

$x+\frac{3}{4} = \frac{\pm \sqrt{-7}}{4}$

$x = \frac{-3 \pm i\sqrt{7}}{4}$

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Question

Use factoring to solve the quadratic equation:

Answer

Factor and solve:

Factor like terms:

Combine like terms:

$x=-\frac{3}{2},\frac{1}{3}$

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Question

Complete the square:

Answer

Begin by dividing the equation by and adding to each side:

$n^2+\frac{1}{2}n = \frac{21}{2}$

Square the value in front of the and add to each side:

$n^2+\frac{1}{2}n + \frac{1}{16} = \frac{21}{2} + \frac{1}{16}$

Factor the left side of the equation:

$(n+\frac{1}{4})^2 = \frac{169}{16}$

Take the square root of both sides and simplify:

$n+\frac{1}{4} = \frac{\pm 13}{4}$

$n = \frac{-1 \pm 13}{4}$

$n=-\frac{7}{2},3$

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Question

Complete the square:

Answer

Begin by dividing the equation by and subtracting $\frac{2}{3}$ from each side:

$x^2+\frac{4}{3}x = -\frac{2}{3}$

Square the value in front of the and add to each side:

$x^2+\frac{4}{3}x +\frac{4}{9} = -\frac{2}{3}+\frac{4}{9}$

Factor the left side of the equation:

$(x+\frac{2}{3})^2 = \frac{-2}{9}$

Take the square root of both sides and simplify:

$x+\frac{2}{3} = \frac{\pm \sqrt{-2}}{3}$

$x = \frac{-2 \pm i\sqrt{2}}{3}$

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Question

Simplify the expression:

$(3x^4y^2)(4xy^2)^{-3}$

Answer

Begin by distributing the exponent through the parentheses. The power rule dictates that an exponent raised to another exponent means that the two exponents are multiplied:

$(3x^{4}y^2)(4xy^2)^{-3}= (3xy^2)(4^{-3}x^{-3}y^{-6})$

Any negative exponents can be converted to positive exponents in the denominator of a fraction:

$\frac{3x^4y^2}{64x^3y^6}$

The like terms can be simplified by subtracting the power of the denominator from the power of the numerator:

$\frac{3x^4y^2}{64x^3y^6}=\frac{3}{64}x^{4-3}y^{2-6}=\frac{3}{64}xy^{-4}$

$\frac{3x}{64y^4}$

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Question

Order the following from least to greatest:

$25^{100}$

$2^{300}$

$3^{500}$

$4^{400}$

$2^{600}$

Answer

In order to solve this problem, each of the answer choices needs to be simplified.

Instead of simplifying completely, make all terms into a form such that they have 100 as the exponent. Then they can be easily compared.

, , , and .

Thus, ordering from least to greatest: .

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Question

What is the largest positive integer, , such that is a factor of ?

Answer

. Thus, is equal to 16.

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Question

$\log_{x}27e^{3}=3$

Solve for .

Answer

First, set up the equation: . Simplifying this result gives .

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Question

Convert the radical to exponential notation.

$\small \sqrt[4]{13}$

Answer

Remember that any term outside the radical will be in the denominator of the exponent.

$\sqrt[b]{x^a}=x^{\frac{a}{b}}$

Since does not have any roots, we are simply raising it to the one-fourth power.

$\sqrt[4]{13}=13^{\frac{1}{4}}$

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Question

What is the value of $9^ \frac{5}{2}$ ?

Answer

An exponent written as a fraction can be rewritten using roots. $9^ \frac{5}{2}$ can be reqritten as $\sqrt[2]{9^5}$ . The bottom number on the fraction becomes the root, and the top becomes the exponent you raise the number to. $\sqrt[2]{9^5}$ is the same as $(\sqrt[2]{9})^{5}=3^{5}$ . This will give us the answer of 243.

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Question

Express the following radical in rational (exponential) form:

$\sqrt{8x^3y^4}$

Answer

To convert the radical to exponent form, begin by converting the integer:

$\sqrt{8x^3y^4}$

$=\sqrt{2^3x^3y^4}$

Now, divide each exponent by to clear the square root:

$=2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^{(\frac{4}{2})}$

Finally, simplify the exponents:

$=2^{(\frac{3}{2})}x^{(\frac{3}{2})}y^2$

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Question

Express the following radical in rational (exponential) form:

$\sqrt[3]{16x^5y^6z^8}$

Answer

To convert the radical to exponent form, begin by converting the integer:

$\sqrt[3]{16x^5y^6z^8}$

$=\sqrt[3]{2^4x^5y^6z^8}$

Now, divide each exponent by to remove the radical:

$=2^{\frac{4}{3}}x^{\frac{5}{3}}y^{\frac{6}{3}}z^{\frac{8}{3}}$

Finally, simplify the exponents:

$=2^{\frac{4}{3}}x^{\frac{5}{3}}y^2z^{\frac{8}{3}}$

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Question

Express the following radical in rational (exponential) form:

$\sqrt[4]{96a^5b^7c^8}$

Answer

To convert the radical to exponent form, begin by converting the integer:

$\sqrt[4]{96a^5b^7c^8}$

$=\sqrt[4]{3\cdot 2^5a^5b^7c^8}$

Now, divide each exponent by to cancel the radical:

$=3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^{\frac{8}{4}}$

Finally, simplify the exponents:

$=3^{\frac{1}{4}}2^{\frac{5}{4}}a^{\frac{5}{4}}b^{\frac{7}{4}}c^2$

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Question

Which fraction is equivalent to $\frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ ?

Answer

Multiply the numerator and denominator by the compliment of the denominator:

$=\frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}} \cdot \frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}$

Simplify the expression:

$=\frac{x^3-x^{\frac{5}{2}}y^\frac{1}{2}}{x-y}$

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Question

Simplify the following radical. Express in rational (exponential) form.

$\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}-x^{\frac{-1}{3}}}$

Answer

Multiply the numerator and denominator by the compliment of the denominator:

$=\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}-x^{\frac{-1}{3}}} \cdot \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

Simplify the expression:

$=\frac{x}{x-1}$

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Question

Choose the fraction equivalent to $\frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ .

Answer

Multiply the numerator and denominator by the compliment of the denominator:

$=\frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}$ $\cdot \frac{x^{\frac{1}{2}}-y^{\frac{1}{2}}}{x^{\frac{1}{2}}-y^{\frac{1}{2}}}$

Simplify the expression:

$=\frac{x-2x^{\frac{1}{2}}y^{\frac{1}{2}}+y}{x-y}$

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Question

Simplify the following radical. Express in rational (exponential) form.

$(\frac{a^{\frac{-2}{3}}}{2^{\frac{1}{2}}a^2})^{\frac{-1}{2}}$

Answer

Multiply the numerator and denominator to the exponent:

$(\frac{a^{\frac{-2}{3}}}{2^{\frac{1}{2}}a^2})^{\frac{-1}{2}}$

$=\frac{a^{\frac{1}{3}}}{2^{\frac{-1}{4}}a^{-1}}$

Simplify the expression by combining like terms:

$=2^{\frac{1}{4}}a^{\frac{4}{3}}$

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Question

Simplify: $(\frac{2^4a}{a^2b^{-1}})^{\frac{-1}{2}}$

Answer

Multiply the numerator and denominator to the exponent:

$(\frac{2^4a}{a^2b^{-1}})^{\frac{-1}{2}}$

$=\frac{2^{-2}a^{\frac{-1}{2}}}{a^{-1}b^{\frac{1}{2}}}$

Simplify the expression by combining like terms:

$=\frac{a^{\frac{1}{2}}}{4b^{\frac{1}{2}}}$

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Question

Express the following exponent in radical form:

$x^{\frac{1}{3}}y^{\frac{1}{2}}z^{\frac{3}{2}}$

Answer

Begin by converting each exponent to have a denominator of :

$x^{\frac{1}{3}}y^{\frac{1}{2}}z^{\frac{3}{2}}$

$=x^{\frac{2}{6}}y^{\frac{3}{6}}z^{\frac{9}{6}}$

Now, rearrange into radical form:

$=\sqrt[6]{x^2y^3z^9}$

Finally, simplify:

$=z\sqrt[6]{x^2y^3z^3}$

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