Simplify the following, giving your answer in index form:
2^{2} \times 2^{2}
4 y^{3} \times 6 y
y^{\frac{4}{3}} \times y^{5}
\dfrac{x^{6}}{4 x^{4}}
\dfrac{u^{ 2 x + 1}}{u^{x}}
2 y^{\frac{3}{5}} \times 2 y^{\frac{2}{5}}
\dfrac{4 n^{3} \times 4 n^{4}}{16}
p^{18} \div p^{8} \div p^{5}
8 b^{\frac{3}{4}} \div 2 b^{\frac{2}{3}}
m^{9} \div m^{5} \times m^{4}
\left( - \dfrac{7}{5} \right)^{m} \times \left( - \dfrac{7}{5} \right)^{n}
Rewrite the following expressions without brackets:
\left(\dfrac{a}{b}\right)^{3}
Complete the statement below:
15 j^{14} \div \left(⬚\right) = 5 j^{7}
Simplify the following, giving your answer in positive index form:
m^{2} \times m^{ - 7 }
\left( 4 m^{ - 10 }\right)^{4}
\left( 4 m^{ - 8 }\right)^{ - 3 }
\dfrac{12 x^{3}}{4 x^{7}}
\dfrac{9 x^{3}}{3 x^{ - 4 }}
\left(\dfrac{a^{3}}{b^{3}}\right)^{ - 5 }
\left(\dfrac{2 h}{3}\right)^{ - 4 }
\dfrac{\left(m^{ - 3 }\right)^{ - 1 } \times \left(m^{4}\right)^{ - 3 }}{m^{3} \times m^{4}}
Express the following in simplest form without negative indices:
\left(\dfrac{a}{b}\right)^{ - 5 }
2 p^{4} q^{ - 2 } \times 5 p^{ - 4 } q^{ - 5 }
\dfrac{5 p^{5} q^{ - 4 }}{40 p^{5} q^{6}}
Express the fraction \dfrac{m}{n^{4}} as a product using negative indices.
Simplify:
a^{0}
\left( 2 \times 13\right)^{0}
\left(a^{0}\right)^{79}
\left( 13 x^{7}\right)^{0} + 13^{0} - 13 h^{0}
\left( 7 m^{0} + 4\right)^{2}
Simplify the following:
\left(w^{3}\right)^{4}
\left( 3 y^{6}\right)^{2}
\left(u^{x + 1}\right)^{3}
\left( 2 y^{4}\right)^{2} \times \left( 2 y^{2}\right)^{3}
\left(\dfrac{1}{b}\right)^{3}
\left( - \dfrac{5 a}{2} \right)^{3}
\left( 4 a^{8}\right)^{\frac{1}{2}}
\dfrac{\left(x^{3}\right)^{2}}{x^{3}}
\left( 2^{3} \div 3^{2}\right)^{3}
\dfrac{3^{ 4 a + 2} \times 3^{1 + 6 a}}{\left(3^{3}\right)^{ 3 a - 1}}
\dfrac{81^{ 7 a - 4} \times 9^{ 3 a + 2}}{27^{3 - 3 a}}
Find the value of a and b in the following equation:
\dfrac{v^{18}}{w^{24}} = \left(\dfrac{v^a}{w^{4}}\right)^b
Write \left(16^{p}\right)^{4} in the form a^b, where a is a prime number.
Write the following in surd form:
Write the following in index form:
\sqrt{x}
\sqrt[6]{x}
\dfrac{1}{\sqrt{x}}
\sqrt[3]{m^{3}}
y^{3} \times \sqrt[3]{y}
Simplify:
Patricia's working out for evaluating 36^{\frac{3}{2}} is shown to the right:
There is an error in her working. In which line did Patricia make an error?
What should she have written in this line?
Hence, evaluate 36^{\frac{3}{2}}.
\begin{aligned}36^{\frac{3}{2}}&=\left(36^{\frac{1}{2}}\right)^3\\ &= \left(\dfrac{\sqrt{36}}{2}\right)^3\\ &= 3^3\\&= 27\end{aligned}
Consider the expression m^{\frac{1}{4}}.
Complete the following working:
m^{\frac{1}{4}} = (m^⬚)^{\frac{1}{2}}
\text{ }= \sqrt{⬚}^{\frac{1}{2}}
Hence, or otherwise, solve the equation m^{\frac{1}{4}} = 8^{\frac{1}{2}}.
Solve the following equation for k:
\sqrt[k]{y} \times \sqrt[k]{y} \times \sqrt[k]{y} = y^{\frac{1}{2}}
Consider the expression m^{5} \times m \sqrt{m}.
Express it in simplest index form.
Express it in surd form.