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 opposite of powers. As 2 squared is 4, then a square root of 4 must be 2. \(2^2 = 4\). Reversing this gives \(\sqrt{4} = 2\).
To find square roots or cube roots, work backwards from square numbers and cube numbers. If you know that \(15^2 = 225\), then you also know that \(\sqrt{225} = 15\). If you know that \(5^3 = 125\), then you also know that \(\sqrt[3]{125} = 5\).
Estimating powers and roots
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 \(3.7^3\).
3.7 is between 3 and 4. \(3^3 = 27\) and \(4^3 = 64\), so the value of \(3.7^3\) will be between 27 and 64, and closer to 64 than 27 because 3.7 is closer to 4 than 3. So an estimate for \(3.7^3\) would be 50 (the actual value is 50.653).
Roots can be estimated by finding the roots of numbers that have integer values above and below the number.
Example
Estimate the value of \(\sqrt{53}\).
The square numbers above and below 53 are \(49 = 7^2\) and \(64 = 8^2\). This means that the value of \(\sqrt{53}\) is between 7 and 8, and closer to 7 because 53 is closer to 49 than it is to 64. So an estimate for \(\sqrt{53}\) is 7.3 (the exact value is 7.280鈥).