1.03 Negative indices
Consider the following expressions:
Identify the base.
Identify the power.
Is 2^{ - 3 } less than or greater than 1?
Complete the following tables:
| 2^{-4} | 2^{-3} | 2^{-2} | 2^{-1} | 2^{0} | 2^{1} | 2^{2} | 2^{3} | 2^{4} |
| \dfrac{1}{2} | 1 | 2 |
| 10^{-4} | 10^{-3} | 10^{-2} | 10^{-1} | 10^{0} | 10^{1} | 10^{2} | 10^{3} | 10^{4} |
| \dfrac{1}{100} | 100 |
| 3^{-4} | 3^{-3} | 3^{-2} | 3^{-1} | 3^{0} | 3^{1} | 3^{2} | 3^{3} | 3^{4} |
| 27 |
Express the following with positive indices:
a^{ - 9 }
\dfrac{1}{a^{ - n }}
\dfrac{a^{ - 9 }}{4}
\dfrac{a^{ - n }}{b^{ - m }}
p^{ - 2 }
3 x^{ - 4 }
7 x^{ - 9 }
p^{-2}q^3
8p^{-3}
2x^{-8}y^3
Express the following without fractions:
Simplify the following, giving your answers with positive indices:
5 y^{9} \times 4 y^{ - 3 }
7 a^{4} \times 4 a^{-2}
5x^4\times \left(-3x^{-8}\right)
3y^{-2}\times 4y^{-3}
2h^{-4}\times 4h^{11}
3y^{-2}\times 2y^{-5}
-4y^2\times \left(-4y^{-5}\right)
\left(5mp\right)^2\times mp^{-2}
\dfrac{9 x^{2}}{3 x^{9}}
\dfrac{15x^3}{5x^7}
Simplify the following, giving your answers with positive indices:
\left( 2 m\right)^{ - 3 }
\left(4m^{-6}\right)^4
\left(3p^{-4}\right)^{-2}
\left( 3 y^{2}\right)^{ - 2 }
\left(\dfrac{y}{4}\right)^{ - 3 }
\left(\dfrac{x^{5}}{y^{4}}\right)^{ - 1 }
\left(\dfrac{x^{7}}{y^{9}}\right)^{ - 4 }
\dfrac{20 x^{3}}{4 x^{ - 2 }}
\dfrac{10 x^{ - 7 }}{2 x^{ - 3 }}
\left(\dfrac{z}{3}\right)^{ - 4 }
\left(\dfrac{p^{3}}{q^{7}}\right)^{ - 1 }
\left(\dfrac{x^{-4}}{y^{-8}}\right)^{ - 2 }
Evaluate the following:
2^{-2} \times 24
2 \times 3^{-2}
10 \div 2^{-1} +3
20 \times 2^{-2}+6
3^{-1} + 4^{-1}
3 \times 4^{-2}+5 \times 2^{-4}
10^2 \times 5^{-2}
\left(7+3^2\right)\times 2^{-2}
Solve the following equations for n:
\dfrac{1}{25} = 5^{n}
\dfrac{1}{8} = 2^{n}
\left( x^{3} y^{ - 5 }\right)^{n} = x^{ - 12 } y^{20}
\left( a^{-5} b^{ 3 }\right)^{n} = a^{ 15 } b^{-9}
Complete the following:
\left(x^4 y^⬚\right)^⬚=\dfrac{y^{12}}{x^{16}}Simplify:
2 y^{6} \times 4 y^{7} \times 4 y^{ - 5 }
6 y^{7} \times 2 y^{ - 5 } \times 5 y^{3}
2 x^{-2} \times 5 x^{ 5 } \times 4x^{-5}
12 x^{-9} \times 2 x^{ 4} \times 3x^{-2}
4 y^{3} \times 3 y^{8} \div 2 y^{ - 1 }
8 y^{10} \div 2 y^{ - 4 } \div y^{3}
40 x^{-2} \div 5 x^{8 } \div 4x^{-9}
14 x^{-8} \div 2 x \times 3x^{6}
Simplify:
\left({3x^{-4}\times 2x^2}\right)^{2}
\left({10x^{3}y^{-4}}\right)^{2}
\left(4x^{3}\right)^{-1}\times\left( y^{-2}\right)^{-3}
\left(10x^3\right)^{3}\times\left( 2x^4y^{3}\right)^{-2}
\left(\dfrac{27x^{-3}y^6}{3x^2y^{-2}}\right)^{2}
\left(\dfrac{18n^3m^7}{3mn^2}\right)^{-2}
Answer the following questions:
What is 0^4 equal to?
Explain why 0^{-4} is undefined.
A student was writing 5a^{-1} without negative indices and wrote 5a^{-1}=\dfrac{1}{5a}. Explain why their working is incorrect, and write the correct answer.