5. Indices
By studying this lesson you will be able to
write a number in index form, as a power having a prime number as the base,
identify powers that have an algebraic symbol as the base,
expand powers that have an algebraic symbol as the base and
find the value of an algebraic expression by substituting positive integers for the unknowns.
Indices
Index notation is used to write a number which is multiplied repeatedly, in a concise way. Let us recall what has been learnt thus far about indices.
When the index is a positive integer, it denotes how many times the number in the base is multiplied by itself.
You have learnt these facts in Grade 6. Do the following exercise to recall what you have learnt thus far about indices.
(1) Write down each of the following products using index notation.
(i) 4 × 4 × 4
(ii) 7 × 7 × 7 × 7
(iii) 2 × 2 × 3 × 3
(iv) 3 × 3 × 5 × 3 × 5
(3) Fill in the blanks in the following table.
(4) Write the number 16
(i) using index notation with base 2.
(ii) using index notation with base 4.
5.1 Expressing a number in index notation with a prime number as the base
Let us write 8 in index notation with a prime number as the base. Let us write 8 as a product of its prime factors.
Do the following to express a number as a product of powers with prime numbers as bases.
👉Start by dividing the number by the smallest prime number which divides it without remainder,
👉Continue dividing the result by the prime numbers which divide it without remainder, in increasing order of the prime numbers, until the answer 1 is obtained.
👉Write the number as a product of powers of these primes, where the index is the number of times division by that prime is done.
(1)
(i) Write 25 in index notation with 5 as the base.
(ii) Write 64 in index notation with 2 as the base.
(iii) Write 81 in index notation with 3 as the base.
(iv) Write 49 in index notation with 7 as the base.
(2) Write each of the following numbers as a product of powers with prime numbers as bases.
(i) 18 (ii) 24 (iii) 45 (iv) 63 (v) 72
5.2 Powers with an algebraic symbol as the base
We have learnt about powers with a number as the base. Let us now consider instances when the base is an algebraic symbol.
When two powers are connected with a multiplication sign, if the bases of both the powers are not numerical values, then it is not necessary to include the multiplication sign.
5.3 Finding the value of a power by substitution
Let us consider expressions in index notation with bases which are unknowns. By substituting values for the unknown bases, the value of an expression in index notation can be found. In this lesson, only positive integers are substituted.