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Multiplication with rational exponents

Express the following fraction in simplest form, only using positive exponents.

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\frac{-5 y^{6}}{\left(-3 y^{-4}\right)^{4}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{\left(2 w^{-4}\right)^{3}}{-3 w^{-3}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{3\left(n^{-4}\right)^{4}}{-15 n^{9}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{\left(3 s^{-1}\right)^{3}}{6 s^{5}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{2 h}{-4\left(h^{-2}\right)^{2}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-2 j^{9}}{-5\left(j^{-1}\right)^{3}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-12 k^{-10}}{3\left(k^{-5}\right)^{4}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-5 p^{-5}}{\left(3 p^{2}\right)^{5}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-3 n^{-3}}{-3\left(n^{5}\right)^{3}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-4\left(a^{-2}\right)^{4}}{-3 a^{10}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-5 z^{-1}}{3\left(z^{-1}\right)^{4}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{\left(-4 v^{3}\right)^{2}}{12 v^{10}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{2 n^{-2}}{-4(n)^{4}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{12 p^{-10}}{\left(-4 p^{-2}\right)^{4}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{\left(-2 r^{-3}\right)^{3}}{10 r^{-2}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{\left(-4 y^{4}\right)^{2}}{-5 y^{3}}

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Express the following fraction in simplest form, only using positive exponents.

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\frac{-2\left(c^{-5}\right)^{4}}{-5 c^{-10}}

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\sqrt[4]{5^{3}} \cdot \sqrt[4]{5^{5}}

is equivalent to

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5^{2}

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5^{\frac{15}{16}}

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25^{\frac{15}{16}}

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25^{2}

\sqrt[6]{3^{5}} \cdot \sqrt[5]{3^{4}}

is equivalent to

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9^{\frac{2}{3}}

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9^{\frac{49}{30}}

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3^{\frac{49}{30}}

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3^{\frac{2}{3}}

\sqrt[3]{2^{2}} \cdot \sqrt[3]{2}

is equivalent to

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4^{\frac{2}{9}}

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2^{\frac{2}{9}}

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2

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4

\sqrt[5]{2^{4}} \cdot \sqrt[5]{2^{6}}

is equivalent to

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2^{2}

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2^{\frac{24}{25}}

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4^{2}

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4^{\frac{24}{25}}

\sqrt[3]{5^{4}} \cdot \sqrt[5]{5^{6}}

is equivalent to

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5^{\frac{8}{5}}

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25^{\frac{8}{5}}

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25^{\frac{38}{15}}

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5^{\frac{38}{15}}

\sqrt[6]{10^{5}} \cdot \sqrt[5]{10^{3}}

is equivalent to

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10^{\frac{43}{30}}

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10^{\frac{1}{2}}

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100^{\frac{43}{30}}

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100^{\frac{1}{2}}

\sqrt[5]{3^{3}} \cdot \sqrt[5]{3^{3}}

is equivalent to

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9^{\frac{6}{5}}

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9^{\frac{9}{25}}

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3^{\frac{9}{25}}

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3^{\frac{6}{5}}

\sqrt[4]{2^{3}} \cdot \sqrt[3]{2^{4}}

is equivalent to

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4

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2^{\frac{25}{12}}

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2

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4^{\frac{25}{12}}

\sqrt[5]{7^{2}} \cdot \sqrt[5]{7^{2}}

is equivalent to

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7^{\frac{4}{5}}

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49^{\frac{4}{5}}

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7^{\frac{4}{25}}

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49^{\frac{4}{25}}

\sqrt[4]{7} \cdot \sqrt[4]{7}

is equivalent to

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49^{\frac{1}{16}}

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49^{\frac{1}{2}}

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7^{\frac{1}{16}}

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7^{\frac{1}{2}}

\sqrt[4]{5^{3}} \cdot \sqrt[5]{5^{4}}

is equivalent to

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25^{\frac{3}{5}}

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5^{\frac{3}{5}}

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5^{\frac{31}{20}}

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25^{\frac{31}{20}}

\sqrt[5]{7^{4}} \cdot \sqrt[3]{7^{2}}

is equivalent to

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7^{\frac{15}{22}}

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7^{\frac{15}{8}}

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7^{\frac{8}{15}}

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7^{\frac{22}{15}}

\sqrt{5} \cdot \sqrt{5^{3}}

is equivalent to

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5^{\frac{1}{2}}

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5^{\frac{3}{4}}

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5^{\frac{4}{3}}

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5^{2}

\sqrt[5]{7^{4}} \cdot \sqrt[6]{7^{5}}

is equivalent to

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7^{\frac{3}{2}}

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7^{\frac{49}{30}}

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7^{\frac{30}{49}}

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7^{\frac{2}{3}}

\sqrt[6]{7} \cdot \sqrt[3]{7^{5}}

is equivalent to

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7^{\frac{6}{11}}

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7^{\frac{11}{6}}

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7^{\frac{18}{5}}

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7^{\frac{5}{18}}

\sqrt[4]{3} \cdot \sqrt{3}

is equivalent to

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3^{\frac{4}{3}}

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3^{8}

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3^{\frac{1}{8}}

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3^{\frac{3}{4}}

\sqrt[6]{2^{5}} \cdot \sqrt[3]{2^{5}}

is equivalent to

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2^{\frac{5}{2}}

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2^{\frac{2}{5}}

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2^{\frac{25}{18}}

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2^{\frac{18}{25}}

\sqrt[3]{2^{4}} \cdot \sqrt[3]{2^{4}}

is equivalent to

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2^{\frac{16}{9}}

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4^{\frac{8}{3}}

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4^{\frac{16}{9}}

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2^{\frac{8}{3}}

\sqrt{7^{3}} \cdot \sqrt[6]{7^{5}}

is equivalent to

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7^{\frac{4}{5}}

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7^{\frac{5}{4}}

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7^{\frac{7}{3}}

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7^{\frac{3}{7}}

\sqrt{3^{5}} \cdot \sqrt{3^{5}}

is equivalent to

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3^{\frac{1}{5}}

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3^{\frac{25}{4}}

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3^{\frac{4}{25}}

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3^{5}

\sqrt[4]{7^{3}} \cdot \sqrt[3]{7}

is equivalent to

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7^{\frac{13}{12}}

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7^{\frac{1}{4}}

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49^{\frac{13}{12}}

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49^{\frac{1}{4}}

\sqrt[6]{5} \cdot \sqrt[5]{5}

is equivalent to

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25^{\frac{11}{30}}

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25^{\frac{1}{30}}

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5^{\frac{1}{30}}

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5^{\frac{11}{30}}

\sqrt[5]{10^{2}} \cdot \sqrt[5]{10^{2}}

is equivalent to

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10^{\frac{4}{25}}

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10^{\frac{25}{4}}

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10^{\frac{5}{4}}

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10^{\frac{4}{5}}

\frac{1}{3 x}+\frac{1}{2 x}=\frac{1}{6}-\frac{1}{x}

A new shopping mall records

150

total shoppers on their first day of business. Each day after that, the number of shoppers is

15\%

more than the number of shoppers the day before.

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Which expression gives the total number of shoppers in the first

n

days of business?

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1

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1.15\left(\frac{1-150^n}{1-150}\right)

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0.85\left(\frac{1-150^n}{1-150}\right)

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150\left(\frac{1-1.15^n}{1-1.15}\right)

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150\left(\frac{1-0.85^n}{1-0.85}\right)

\frac{d y}{d x}=\frac{15}{x^{4}} \cdot \frac{4 y^{\frac{3}{4}}}{3}

\frac{2 x y^{-2}}{\left(x^{-1}\right)^{-4} \cdot y^{0}}

Simplify the complex rational expression by using the LCD. [Be sure to remove any common factors.]

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\frac{\frac{7}{y}+\frac{y}{r}}{\frac{r}{y}-\frac{9}{r}}

(b d)^{\frac{1}{4}} \cdot\left(\frac{b}{d}\right)^{4}

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Which of the following expressions is equivalent to the given expression assuming

d

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1

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b^{\frac{17}{4}} d^{\frac{17}{4}}

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b^{\frac{17}{4}} d^{\frac{15}{4}}

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b^{2}

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\frac{b}{d}

2 \frac{1}{2}-1 \frac{3}{4}

\frac{7}{10}+\frac{2}{10}-\frac{8}{10}

6^{\frac{1}{6}} \cdot 2^{\frac{5}{6}} \cdot 3^{\frac{7}{6}}

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Which of the following is equivalent to the given expression?

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1

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6 \sqrt[3]{3}

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2^{\frac{5}{36}} \cdot 3^{\frac{7}{36}}

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36^{\frac{35}{216}}

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\sqrt[3]{6^{13}}

Solve the equation using the multiplication property.

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\frac{2}{15}=-\frac{y}{5}

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The solution set is

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