Powers and roots - Higher

Powers, or 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
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The small floating digit is known as the power or index number.
Example of how to write numbers in index form. Base number shown as 2 and the power or index number shown as 4.

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 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.

3.7 is between 3 and 4. 3 to the power of 3 = 27 and 43 = 64, so the value of 3.7 to the power of 3 will be between 27 and 64, closer to 64 than 27 because 3.7 is closer to 4 than 3.

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 √53 lies between 7 and 8

Prime factors

Prime factors are any factors 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 integer has a unique prime factorisation, so it doesn’t 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 factor pairs. Break down each number until you have found prime numbers. Then circle these primes.

Factors of the number 40

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

A prime factor tree showing 24 as a product of it prime factors, 12, 6 and 3.

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 .