Indices
- Ten to the power of five can be written as \(10^5\)It means \(10 \times 10 \times 10 \times 10 \times 10 = 100,000\)
- \(2^3 = 2 \times 2 \times 2 = 8\)
Terms used for indices
Examples of indices
Calculate 5虏
Answer
\( 5^2 = 5 \times 5 = 25 \)
Find the value of 4鈦
Answer
\(4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024\)
Calculating with indices
Any calculations must follow the correct order of operations:
Brackets
Indices
Division
Multiplication
Addition
Subtraction
Calculate the value of 2鲁 x 3虏
Following order of operation, the indices are calculated first before multiplying.
\(2^3 \times 3^2 = 8 \times 9 = \color{red} \textbf {72}\)
Examples of calculating with indices
Calculate 6虏 - 2鈦
Answer:\(6^2 - 2^4 = 36 鈥 16 = \color {red} \textbf {20}\)
Find the value of 4鲁 梅 2鈦
Answer:\(4^3 - 2^5 = 64 \div 32 = \color {red} \textbf {2}\)
Question
What is the value of 3虏 x 2虏?
Solution:
Calculate the indices
\(3^2 \times 2^2 = 9 \times 4\)
Solution:
9 x 4 = 36
Question
Evaluate 4虏 +3鲁
Solution:
16 + 27 = 43
Rules of indices when working with the same base
- Rule 1 - When multiplying, ADD the indices
- Rule 2 - When dividing, SUBTRACT the indices
- Rule 3 - When the power is raised to another power, MULTIPLY the indices.
- Rule 4 - Anything to the power of zero is 1
Rule 1 - multiplying
To find the value of\(\color{red} \hspace{2em} 4^2 \hspace{1.45em} \times \hspace{2.75em} 4^3\)
It can be rewritten as \(\hspace{0.25em} 4 \times 4 \quad \times \quad 4 \times 4 \times 4=4^5\)
So \(4^2 \times 4^3 =4^{\color{red}{(2+3)}} =4^5 =\color{red}\textbf {1024}\)
When multiplying the rule is to add the indices.
Rule 2 - dividing
To find the value of\(\color{red} \hspace{6.35em} 5^7 \hspace{5.65em} \div \hspace{5em} 5^4 \)
It can be rewritten as \(\hspace{0.25em} 5 \times 5 \times 5\times 5\times 5\times 5\times 5\quad \div \quad \times 5 \times 5 \times 5 \times 5\)
\( = \hspace{12em}78125 \hspace{5.2em} \div \hspace{4.8em}625\)
\( =\hspace{12em} \color{red}\textbf {125}\)
A quicker way
\(5^7 \div 5^4 = 5^{(7-4)} = 5^3 = \color{red}\textbf {125} \)
When dividing the rule is to subtract the indices.
Rule 3 - raising to a power
Simplify \(\color{red} (3^3)^2 \)
It can be rewritten as \(3^3 \times 3^3 \)
And using Rule 1
\(3^3 \times 3^3 = 3^{3+3} = 3^6 = \color{red}\textbf {729}\)
When raising a power to another power multiply the indices
Rule 4 - raising to the power of 0
\(6^2 \div 6^2\)
using Rule 2
\(6^2 \div 6^2 = 6^{(2-2)} = 6^0 = \color{red}\textbf {1}\text { since } 6^2 \div 6^2 \text{ and } 36 \div 36 = 1\)
\(9^0 = 1 \hspace{5em} m^0 = 1 \hspace{5em}725^0 = 1\)
Anything to the power of zero is 1
Question - rule 1
Simplify 7鈦 x 7鈦 giving the answer in index form
Answer:
\(7^7 \times 7^4 = 7^{7+4} = 7^{11}\)
Question - rule 2
Evaluate 2鈦 梅 2鈦
Answer:
\(2^8 \div 2^5 = 2^{8-5} = 2^3 = 8\)
Question - rule 3
Simplify (a鲁)虏
Answer:
\(a^{3 \times 2} = a^6\)
Question - rule 4
Evaluate 5鈦
Answer:
\(5^0 = 1\)
Question
Simplify 6虏 x 6鲁 giving your answer in index form
Answer:
\(6^{2+3} = 6^5\)
Question
Simplify 5鲁 梅 5鲁 giving your answer in index form
Answer:
\(5^{3-3} = 5^0 = 1\)
Test yourself
More on M2: Number
Find out more by working through a topic
- count5 of 6
- count6 of 6