Understanding Powers and Roots
A power represents repeated multiplication of the same number. For example, squaring a number means multiplying it by itself once: \(a^2 = a \times a\). Cubing a number means multiplying it by itself twice: \(a^3 = a \times a \times a\).
Roots are the inverse operations of powers. The square root of a number \(x\) is the number which, when squared, gives \(x\). Similarly, the cube root of \(x\) is the number which, when cubed, equals \(x\).
For example, since \(5^2 = 25\), the square root of 25 is 5, written as \(\sqrt{25} = 5\). Since \(3^3 = 27\), the cube root of 27 is 3, written as \(\sqrt[3]{27} = 3\).
- Powers are repeated multiplication.
- Roots reverse the effect of powers.
- Square root and cube root are specific types of roots.
Squares and Square Roots of Numbers 1 to 15
Memorising squares of numbers from 1 to 15 helps in quick calculations and recognising perfect squares. The squares are:
\[1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36, 7^2=49, 8^2=64, 9^2=81, 10^2=100, 11^2=121, 12^2=144, 13^2=169, 14^2=196, 15^2=225\]
The square root of these numbers is the original integer from 1 to 15. For example, \(\sqrt{169} = 13\) because \(13^2 = 169\).
- Squares of 1 to 15 are essential to memorise.
- Square roots of perfect squares are integers 1 to 15.
- Use these to identify or verify square roots quickly.
Cubes and Cube Roots of Numbers 1, 2, 3, 4, 5, and 10
Cubes involve multiplying a number by itself twice. The cubes of 1, 2, 3, 4, 5, and 10 are:
\[1^3=1, 2^3=8, 3^3=27, 4^3=64, 5^3=125, 10^3=1000\]
The cube root reverses this operation. For example, \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
- Know cubes of 1, 2, 3, 4, 5, and 10.
- Cube roots give the original number when cubed.
- Useful for solving problems involving volume or powers of three.
Applying Powers and Roots in Calculations
To calculate powers, multiply the base number by itself the required number of times. For example, to find \(5^2\):
\[5^2 = 5 \times 5 = 25\] To find the square root of a number, identify the integer which squared equals the number. For example, to find \(\sqrt{169}\):
\[\sqrt{169} = 13 \quad \text{since} \quad 13^2 = 169\] For cube roots, find the number which cubed equals the given number. For example, to find \(\sqrt[3]{8}\): \[\sqrt[3]{8} = 2 \quad \text{since} \quad 2^3 = 8\] If the number is not a perfect square or cube, approximate or use a calculator as needed.
- Calculate powers by repeated multiplication.
- Find roots by identifying the inverse operation.
- Check answers by reversing the operation.
Examples
1. Find the square root of 196. Since \(14^2 = 196\), \(\sqrt{196} = 14\).
2. Write down the value of \(5^2\). \(5^2 = 5 \times 5 = 25\).
3. Work out the cube root of 27. \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
- Use known squares and cubes to solve problems quickly.
- Verify results by squaring or cubing the root found.
- Apply these skills to more complex calculations involving powers and roots.
At-a-glance summary
Comparison table
| Step | Detail | Example | Exam Use |
|---|---|---|---|
| Recall squares | Memorise squares of 1 to 15 | \(12^2 = 144\) | Quickly identify perfect squares in questions |
| Recall cubes | Memorise cubes of 1,2,3,4,5,10 | \(4^3 = 64\) | Solve volume or cube root problems efficiently |
| Calculate powers | Multiply base repeatedly | \(5^2 = 5 \times 5 = 25\) | Show clear working for marks |
| Calculate roots | Find inverse of power | \(\sqrt{169} = 13\) | Use to simplify expressions or solve equations |
| Check answers | Reverse operation to verify | Check \(\sqrt{25} = 5\) by \(5^2 = 25\) | Avoid careless errors in exams |
Active recall
Quick check
1. What is the square of 7?
\(7^2 = 7 \times 7 = 49\)
2. Find the cube root of 125.
\(\sqrt[3]{125} = 5\) because \(5^3 = 125\)
3. Write down the square root of 81.
\(\sqrt{81} = 9\) since \(9^2 = 81\)
4. Calculate \(3^3\).
\(3^3 = 3 \times 3 \times 3 = 27\)
5. Is 50 a perfect square?
No, because there is no integer \(n\) such that \(n^2 = 50\).