Powers and roots - Higher
Powers, or Shows how many times a number has been multiplied by itself. The plural of index is indices., are ways of writing numbers that have been multiplied by themselves:
- \(2 \times 2\) can be written as 22 (2 squared)
- \(2 \times 2 \times 2\) can be written as 23 (2 cubed)
- \(2 \times 2 \times 2 \times 2\) can be written as 24 (2 to the power of 4), and so on
Roots
Roots are the reverse of powers. For example, a square root is the reverse of a square number.
So \(5^2 = 25\) and \(\sqrt{25} = 5\)
A cube root is the reverse of a cube number.
So \(5^3 = 125\) and \({3}\sqrt{125} = 5\)
Estimating powers and roots - higher
Powers of any number can be estimated by finding the nearest Integers are whole numbers. above and below the number.
Example
Estimate the value of 3.73
If \(3^3 = 27\) and \(4^3 = 64\)
3.73 must be between 27 and 64
A good estimate is that 3.73 is about 50 but other answers would also be acceptable.
Roots can be estimated by finding the roots of numbers that have integer values above and below the number.
Example
Between which integer values does the square root of 53 lie?
\(7^2 = 49 \) and \(8^2 = 64\) so 鈭53 lies between 7 and 8
Prime factors
A number that only has two factors - itself and one. factors are any A factor is a number which divides exactly into another number. 1 is a factor of every number and every number is a factor of itself. A number can have several factors. Example: 1, 2, 5 and 10 are the factors of 10. of a number that are also prime numbers. There are many methods to find the prime factors of a number, but one of the most common is to use a prime factor tree.
Example
Write 40 as a product of its prime factors.
Firstly, find two numbers that will multiply together to give 40. For example \(4 \times 10 = 40\) would be one way of doing this calculation. Every Integers are whole numbers. has a unique prime factorisation, so it doesn鈥檛 matter which factors are chosen to start the factor tree as you will end up with the same answer.
Neither 4 nor 10 is a prime number, and this question is looking for prime factors, so each number must be broken down again into Pairs of factors multiplied together to get a certain product.. Break down each number until you have found prime numbers. Then circle these primes.
The question has asked for a product of prime factors. To find a product, we have to multiply, so take all the prime numbers found in the prime factor tree and multiply them together.
This gives \(2 \times 2 \times 2 \times 5\). This can be written in index form as \(2^3 \times 5\)
This answer can be checked by making sure \(2 \times 2 \times 2 \times 5\) is equal to 40. \(2 \times 2 \times 2 \times 5 = 40\), so this answer is correct. The final answer is \(2^3 \times 5\).
Question
Express 24 as a product of prime factors.
Start by looking for a factor pair of 24.
\(2 \times 12 = 24\)
Continue until all the factors are prime numbers. Here is one way to complete the factor tree
When all primes have been found and circled, check that the answer multiplies to 24. \(2 \times 2 \times 2 \times 3 = 24\), so the product of prime factors is \(2 \times 2 \times 2 \times 3 = 2^3 \times 3\) .