Key points
Squares, cubes and higher powers are shown as small digits called Indices are powers eg, 3 to the power of 2, written 3² (sometimes these are called How many times to use the number in a multiplication. For example, 3² is 3 to the power of 2 or 3 squared. or The exponent of a number says how many times to use the number in a multiplication. Also known as index, a number, positioned above and to the right of another (the base), indicating repeated multiplication when the exponent is a positive integer.). Indices is the plural of Positioned above and to the right of a number. It is an abbreviation of repeated multiplication. Eg, 7³ means 7 x 7 x 7.
Laws of indices provide us with rules for simplifying calculations or expressions involving powers that have the same The number that gets multiplied when using an exponent (index).. This means that the larger number or letter must be the same.
The opposite of The square of a number is the product of the number and itself. n2 is n squared or n to the power of two. For example, the square of 2 is 2² = 2 × 2 = 4 is A square root of a number is a value that, when multiplied by itself, gives the number. Eg, 4 × 4 = 16, so the square root of 16 is 4.
The opposite of The result of multiplying to power of three, n3 is n cubed or n to the power of three. For example, the cube of 2 is 2³ = 2 × 2 × 2 = 8 is The cube root of a number is a special value that, when used in a multiplication three times, gives that number. Eg, 3 × 3 × 3 = 27, so the cube root of 27 is 3.
Index laws for multiplication
- When multiplying Indices are powers eg, 3 to the power of 2, written 3² it’s important to understand index notation. For example, 7 × 7 × 7 × 7 can be recorded as 7⁴
- The Index laws are the rules for simplifying expressions involving powers of the same base number. for multiplication only works if the The number that gets multiplied when using an exponent (index). numbers are the same. Eg, in the calculation 2² x 2³ the base numbers are the same (2).
- When multiplying indices, if the base values are the same, the expression can be simplified by adding the indices.
- The index law for multiplying still applies even when indices are negative.
Examples
Image caption, Using index notation, simplify 7² x 7³
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Question
Find the value of the missing index value.
The base values are the same (y).
Add the indices together (6 + 5 = 11)
y⁶ x y⁵ = y¹¹
The missing index value is 11
Index laws for division
- When dividing Indices are powers eg, 3 to the power of 2, written 3² , it’s important to understand index notation. For example, a × a × a × a × a can be recorded as a⁵
- The Index laws are the rules for simplifying expressions involving powers of the same base number. for division only works if the The number that gets multiplied when using an exponent (index). numbers are the same.
- When dividing indices, if the base values are the same, the expression can be simplified by subtracting the indices.
- Remember that division can be written as a fraction, for example 7 ÷ 2 can be written as \( \frac{7}{2} \)
Examples
Image caption, Using index notation, simplify 8⁵ ÷ 8³
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Question
Work out the value of the missing index value.
The base values are the same (x).
Subtract the indices (7 – 3 = 4)
x⁷ ÷ x³ = x⁴
The missing index value is 4
Practise laws of indices
Practise what you have learned about the laws of indices with this quiz.
Quiz
Game - Divided Islands
Play the Divided Islands game! game
Using your maths skills, help to build bridges and bring light back to the islands in this free game from BBC Bitesize.
