Mathematical Relationships and Basic Graphs - High School Math

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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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Question

Express the following exponent in radical form:

$r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$

Answer

Begin by converting each exponent to have a denominator of :

$r^2s^{\frac{1}{3}}y^{\frac{1}{2}}$

$=r^{\frac{12}{6}}s^{\frac{2}{6}}y^{\frac{3}{6}}$

Now, put this in radical form:

$=\sqrt[6]{r^{12}s^2y^3}$

Finally, simplify:

$=r^2\sqrt[6]{s^2y^3}$

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Question

Express the following exponent in radical form:

$(3x)^{\frac{1}{2}}x^{\frac{1}{4}}$

Answer

Begin by changing the fractional exponents so that they both have a common denominator of :

$(3x)^{\frac{1}{2}}x^{\frac{1}{4}}$

$=(3x)^{\frac{2}{4}}x^{\frac{1}{4}}$

Now, put this in radical form and simplify:

$=\sqrt[4]{(3x)^2x}$

$=\sqrt[4]{9x^2x}$

$=\sqrt[4]{9x^3}$

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Question

Simplify the following radical expression using exponents. Express the final answer in radical form.

$\sqrt[6]{8n^9}$

Answer

Begin by converting the radical into exponent form:

$\sqrt[6]{8n^9}$

$(8^{\frac{1}{6}}n^{\frac{9}{6}})$

Simplify the exponent and multiply:

$(2^3^{\frac{1}{6}}n^{\frac{9}{6}})$

$(2^{\frac{1}{2}}n^{\frac{3}{2}})$

Convert into radical form:

$\sqrt{2n^3}$

Simplify:

$n\sqrt{2n}$

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Question

Find the value of $\sqrt{80} + \sqrt{20}$ .

Answer

To solve this equation, we have to factor our radicals. We do this by finding numbers that multiply to give us the number within the radical.

$\sqrt{80} = \sqrt{20}\sqrt{4}$

$\sqrt{20} = \sqrt{5}\sqrt{4}$

Add them together:

$\sqrt5\sqrt4\sqrt4 + \sqrt5\sqrt4$

4 is a perfect square, so we can find the root:

$(2)(2)\sqrt5 + 2\sqrt5$

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

Since both have the same radical, we can combine them:

$4\sqrt5 + 2\sqrt5 = 6\sqrt5$

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Question

Simplify the expression:

$\frac{3\sqrt[4]{32}}{2\sqrt[4]{162}}$

Answer

Use the multiplication property of radicals to split the fourth roots as follows:

$\rightarrow \frac{3\sqrt[4]{16}\sqrt[4]{2}}{2\sqrt[4]{81}\sqrt[4]{2}}$

Simplify the new roots:

$\rightarrow \frac{3(2)\sqrt[4]{2}}{2(3)\sqrt[4]{2}}$

$\rightarrow \frac{6\sqrt[4]{2}}{6\sqrt[4]{2}}$

$\rightarrow 1$

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