Solve the following equations:
x^{5} = 3^{5}
8^{ - 7 } = x^{ - 7 }
x^{3} = \left(\dfrac{8}{5}\right)^{3}
x^{ - 7 } = \dfrac{1}{6^{7}}
3 \left(x^{ - 9 }\right) = \dfrac{3}{2^{9}}
x^{\frac{1}{3}} = \sqrt[3]{6}
\sqrt[3]{5} = x^{\frac{1}{3}}
\dfrac{1}{2^{5}} = x^{-5}
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:
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
Find the missing values in the following equations:
Solve the following equations:
Find the interval, of two consecutive integers, in which the solution of the following equations will lie:
Consider the following equations:
Rewrite each side of the equation with a base of 2.
Hence, solve for x.
8^{x} = 4
16^{x} = \dfrac{1}{2}
\dfrac{1}{1024} = 4^x
\left(\sqrt{2}\right)^{x} = \sqrt[5]{32}
Solve the following exponential equations:
9^{y} = 27
\left(\sqrt{6}\right)^{y} = 36
\left(\sqrt{2}\right)^{k} = 0.5
3^{ 5 x - 10} = 1
\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}
\left(\dfrac{1}{9}\right)^{x + 5} = 81
\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}
\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}
8^{x + 5} = \dfrac{1}{32 \sqrt{2}}
30 \times 2^{x - 6} = 15
2^{x} \times 2^{x + 3} = 32
81^{x - 1} = 9^{ 3 x + 5}
25^{x + 1} = 125^{ 3 x - 4}
a^{x + 1} = a^{3} \sqrt{a}
3^{x^{2} - 3 x} = 81
27 \left(2^{x}\right) = 6^{x}
For each of the following equations:
Simplify the left side of the equation.
Solve the equation for x.
Consider the following equations:
Determine the substitution, m that would reduce the equation to a quadratic.
Hence, solve the equation for x.
Given that 2^{x} \times 5^{x + 2} = 50, find the value of 2^{x + 2} \times 5^{x}.
Find the x-coordinate of the point of intersection of the graphs of y = 2^{ 5 x} and y = 4^{x - 3}.
Find the value of h, given the point \left(h,\dfrac{1}{9}\right) lies on the curve y = 3^{ - x }.
Given the points \left(3, n\right), \left(k, 16\right) and \left(m, \dfrac{1}{4}\right) all lie on the curve with equation y = 2^{x}, find the value of:
n
k
m
A certain type of cell splits in two every hour and each cell produced also splits in two each hour. The total number of cells after t hours is given by:
N(t)=2^tFind the time when the number of cells will reach the following amounts:
1024
4096
The frequency f \left(\text{Hz}\right) of the nth key of an 88-key piano is given by f \left( n \right) = 440 \left(2^{\frac{1}{12}}\right)^{n - 49}.
Find the frequency of the forty-ninth key.
Find the frequency of the 40th key to the nearest whole number.
Find the value of n that corresponds to the key with a frequency of 1760 \text{ Hz}.
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.