Write the following in surd form:
Write the following as a single power of 2:
Write the following as a single power of 3:
Complete the following statement: \begin{aligned} \sqrt{m^{8}} &=\left(m^{8}\right)^{⬚}\\ &=m^{ ⬚ \times \frac{1}{2}}\\ &=m^{⬚} \end{aligned}
Write the following in the form x^k, where k is rational:
\sqrt{x}
\sqrt[6]{x}
\dfrac{1}{\sqrt{x}}
Consider the expression 64^{\frac{2}{3}}.
Complete the statement: 64^{\frac{2}{3}} = \left(\sqrt[⬚]{64}\right)^⬚
Hence, evaluate 64^{\frac{2}{3}}.
Without using a calculator, evaluate the following:
Use a calculator to evaluate the following, to two decimal places:
Simplify the following expressions, giving your answers in index form. Assume that all variables represent positive numbers.
Simplify the following expressions, giving your answers in surd form. Assume that all variables represent positive numbers.
Determine whether the following statements accurately describe the meaning of the expression x^{ - \frac{y}{z} }:
x^{ - \frac{y}{z} } means we are raising x to the power of \dfrac{z}{y}, then taking the reciprocal of the result.
x^{ - \frac{y}{z} } means we are taking the reciprocal of x, then raising the result to the power of \dfrac{y}{z}.
x^{ - \frac{y}{z} } means we are taking the reciprocal of x, then raising the result to the power of \dfrac{z}{y}.
x^{ - \frac{y}{z} } means we are raising x to the power of \dfrac{y}{z}, then taking the reciprocal of the result.
Describe how we could interpret the expression m^{\frac{q}{r}} in terms of powers and roots of m.
Is there a real number that equals \sqrt[4]{ - 16 }? Explain your answer.
Consider the expression m^{5} \times m \sqrt{m}.
Express it in simplest index form.
Express it in surd form.
Solve the following equation for k:
\sqrt[k]{y} \times \sqrt[k]{y} \times \sqrt[k]{y} = y^{\frac{1}{2}}
To evaluate 81^{\frac{3}{2}} would it be more efficient to use the property a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^{m}, or the property a^{\frac{m}{n}} = \sqrt[n]{a^{m}}. Explain your answer.
Simplify the following, giving your answers with positive indices. Assume that all variables represent positive numbers.
\dfrac{2}{m^{5}} \times \dfrac{m^{6}}{5}
\dfrac{4}{m^{4}} \times \dfrac{m^{2}}{5}
\dfrac{3}{m^{\frac{5}{9}}} \times \dfrac{m^{\frac{7}{9}}}{4}
\dfrac{2}{m^{\frac{7}{9}}} \times \dfrac{m^{\frac{3}{9}}}{3}
\dfrac{3}{\sqrt{m^{7}}} \times \dfrac{\sqrt{m^{3}}}{4}
\dfrac{\sqrt[3]{m^{5}}}{4} \times \dfrac{3}{\sqrt[3]{m^{7}}}
\dfrac{\sqrt{m^{7}}}{\sqrt[4]{m^{5}}} \times \dfrac{\sqrt{m^{9}}}{\sqrt[4]{m^{3}}}
\dfrac{3}{m^{\frac{2}{7}}} \div \dfrac{4}{m^{\frac{5}{7}}}
\dfrac{m^{\frac{7}{3}}}{m^{\frac{3}{2}}} \div \dfrac{m^{\frac{1}{2}}}{m^{\frac{8}{3}}}
\dfrac{m^{\frac{1}{3}}}{m^{\frac{11}{2}}} \div \dfrac{m^{\frac{7}{2}}}{m^{\frac{8}{3}}}
\dfrac{m^{5}}{m^{3}} \div \dfrac{m^{2}}{m^{7}}
\dfrac{m^{2}}{m^{5}} \div \dfrac{m^{4}}{m^{3}}
\dfrac{x^{\frac{6}{7}} + x^{\frac{5}{7}}}{x^{\frac{4}{7}}}
\dfrac{x^{\frac{3}{5}} + x^{\frac{2}{5}}}{x^{\frac{4}{5}}}
\dfrac{x^{\frac{8}{9}} - x^{\frac{7}{9}}}{x^{\frac{2}{9}} \times x^{\frac{3}{9}}}
\dfrac{x^{\frac{8}{9}} - x^{\frac{5}{9}}}{x^{\frac{4}{9}} \times x^{\frac{3}{9}}}
Simplify the following, giving your answers in surd form with positive indices. Assume that all variables represent positive numbers.
\dfrac{2}{\sqrt{m^{4}}} \div \dfrac{\sqrt{m^{7}}}{5}
\dfrac{\sqrt[3]{m^{2}}}{2} \div \dfrac{5}{\sqrt[3]{m^{5}}}
Simplify the following: