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.
Describe how we could interpret the expression m^{\frac{q}{r}} in terms of powers and roots of m.
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.
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.
Consider the expression m^{\frac{1}{4}} = 8^{\frac{1}{2}}.
Complete the following working:
\begin{aligned} m^{\frac{1}{4}} &= (m^⬚)^{\frac{1}{2}}\\ &= \sqrt{⬚}^{\frac{1}{2}} \end{aligned}
Hence, or otherwise, solve the equation m^{\frac{1}{4}} = 8^{\frac{1}{2}}.
Consider the equation x^{\frac{2}{3}} = 9.
Complete the statement: \left(x^{\frac{2}{3}}\right)^{⬚} = x
Hence, solve the equation x^{\frac{2}{3}} = 9.
Find the missing values in the following equations:
Without using a calculator, evaluate the following:
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.
Solve the following equations for x:
x^{\frac{3}{4}} = 125
\left(x - 4\right)^{0.5} = 4
\left(2x - 1\right)^{\frac{2}{3}} = 9
\left( 4 x + 3\right)^{ - 1.5 } = 8
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.
The surface area A of a cube with a volume V is given by A = 6 V^{\frac{2}{3}}.
Find the surface area of a cube with a volume of 8 \text{ cm}^{3}.
Find the volume of a cube with a surface area of 216 \text{ cm}^{2}.
The volume V of a sphere with surface area A is given by: V = \dfrac{A^{\frac{3}{2}}}{6 \sqrt{\pi}}
Find the volume of a sphere with a surface area of 50 \text{ cm}^{2}. Round you answer to the nearest cubic centimetre.
Find the surface area of a sphere with a volume of 125 \text{ cm}^{3}. Round you answer to the nearest square centimetre.
The volume V of a regular tetrahedron with edge length a is given by: V = \frac{a^3}{6\sqrt{2}}
Find the volume of a tetrahedron with a side length of 2 \text{ cm}.
Rearrange the formula to make a the subject.
Hence, find the side length of a tetrahedon with a volume of 72 \text{ cm}^3.
A company's net profit (in dollars) can be modelled by the equation: P = 50 n^{\frac{3}{4}} - 2500 Where n is the number of units sold.
Find its net profit be in dollars if it manages to sell 256 units.
Find the number of units needed to be sold to make a net profit of \$8300.
A company's net profit (in dollars) can be modelled by the equation: P = 180 n^{\frac{3}{5}} - 4000 Where n is the number of units sold.
Find its net profit be in dollars if it manages to sell 32 units.
Find the number of units needed to be sold to make a net profit of \$45\,000. Round to the nearest whole unit.