We know that 7² = 49. We can also express this in the following way:

We say that 7 is equal to the square root of 49. In general, the square root of a number a is defined as another number b where b² = a.

In the same way, we can define the n^th root of a number a as b where bⁿ = a

and we write:

It is important to be aware that not all numbers have roots. For example, -4 does not have a square root as the square of any number, whether it be positive or negative, is always positive. For this reason no negative number has a square root and no negative number has a root with an even index.

15. Work out the following referring back to the list you made of perfect squares and cubes:

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Thinking back to the previous example we can safely say that:

16. Express the following in root form. Work out the value of the power. Use the following window to check your answers. Increase the number of decimal places if necessary.

a) 16^3/4   b) 27^2/3    c) 125^4/3

d) 64^5/6   e) 100^-3/2   f) 8^-2/3

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Numbers raised to fractional indices have the same properties as those raised to the power of a whole number. Let's go over them again one by one:

The product of powers with the same base number.

The product of two powers with the same base number is the same base number whose index is the sum of the other two indices.

a^m * aⁿ = a^m+n

This rule is true for any base number or index, whether the number be positive, negative, a whole number or a fraction.

17. Work out the answers to the following products in your notebook and write them in index form:

a) 2^3/5 * 2^7/2
b) 3^5/2 * 3^2/3
c) 5^2/5 * 5^2/3
d) 2^-3/10 * 2^2/5
e) 3^-5/2 * 3^-2/3
f) 10^-1/5 * 10^1/3

Check your results in the following window.

Dividing powers with the same base number.

Similarly to the product rule, the following general rule applies to both positive and negative indices:

Dividing two powers with the same base number gives the same base number whose index is the difference between the other two indices.

a^m : aⁿ = a^m-n

18. Work out the following divisions in your notebook and write them in index form:

a) 2^7/3 : 2^4/3
b) 3^1/5 : 3^2/3
c) 5^1/6 : 5^1/3
d) 64^3/2 : 64^-1/3
e) 3^-1/2 : 3^3/2
f) 8^-4/3 : 8^-5/3

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A product raised to a power.

A product raised to a power is equal to the product of the base numbers raised to the same power.

(a*b)^m = a^m * b^m

19. Express the following in product form:

a) (2*5)^1/6
b) (3*4)^3/2
c) (2*8)^2/3
d) (4*6)^3/4
e) (2*5)^-1/2
f) (3*2)^-2/3
g) (2*5)^-5/3

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Dividing numbers raised to a power.

This works in a similar way to the product rule.

A division raised to a power is equal to one number raised to a power divided by another number raised to the same power.

(a/b)^m = a^m / b^m

20. Express the following in division form:

a) (18/2)^5/6
b) (64/4)^1/2
c) (75/5)^2/3
d) (12/3)^3/4
e) (18/2)^-2/3
f) (32/4)^-3/2
g) (81/27)^-1/3
h) (32/9)^-1/4

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A power raised to a power.

A power raised to another power is the same as the base number raised to the product of these two powers:

(a^m)ⁿ = a^m*n

21. Express the following in your notebook as numbers raised to just one power:

a) (2^1/3)⁷
b) (3⁵)^1/3
c) (5^1/5)^1/3
d) (2^-3/2)⁴
e) (3^3/4)^-1/4
f) (25^-1/2)^-3

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22. Use the rules in the examples above to help you to write a list of all the rules of roots and their operations.