Law of multiplication of indices

a^m x aⁿ = a^{m + m}

Law of division of indices

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

a⁰ = 1

$\frac{5^{7}x{5}^{-4}}{5^3}$ = $\frac{5^{\mathrm{7+-4}}}{5^3}$

$\frac{5^{7}x{5}^{-4}}{5^3}$ = $\frac{5^{\mathrm{7-4}}}{5^3}$

$\frac{5^{7}x{5}^{-4}}{5^3}$ = $\frac{5^3}{5^3}$ = 1

Note: 5³ divides 5³ once.

Alternatively, the law of division could be applied.

$\frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}}$ = a^{m - n}

$\frac{5^3}{5^3}$ = 5^3-3

$\frac{5^3}{5^3}$ = 5⁰ = 1

5⁰ = 1 (a⁰ = 1)

Convert the 81 to 3^y, where y is the power that will produce 81

81 = 3 x 3 x 3 x 3

From the law of indices, a^b x a^c = a^b+c

81 = 3 x 3 x 3 x 3 = 3^1+1+1+1 = 3⁴

Now replace 81 by 3⁴ in the equation

3^x = 81

3^x = 3⁴

Since the bases are the same, the powers are also the same

Hence x = 4

Applying the indices principle

The expression can therefore be rewritten as:

Cancel out

Concepts

1. a^x x a^y = a^x+y, thus if the bases are the same and multiplying, add the powers

2. (a^xx b^y)ⁿ = a^{x x n} x b^{y x n}, thus the outside power multiplies all the powers inside

Applying these two concepts,

(2ab²)² = 2^1x2a^1x2b^2x2 = 2²a²b⁴ = 4a²b⁴

NOTE: Every number or letter is raised to the power 1, which has no effect, hence 2 = 2¹ and a = a¹

⇒ (2ab²)² x 3a³b = 4a²b⁴ x 3a³b

Simplifying the numbers and the ones with the same bases

⇒ 4a²b⁴ x 3a³b = 4 x 3 x a²xa³b⁴xb

Applying concept 1,a^x x a^y = a^x+y

⇒ 4 x 3 x a²xa³b⁴xb = 12a²⁺³b⁴⁺¹ = 12a⁵b⁵

NOTE: b = b¹ ⇒ b⁴xb = b⁴xb¹ = b⁴⁺¹ = b⁵

Make sure both sides of the equation are in the same base

4 and 32 can both be expressed in base 2

Applying the law of indices, a^b x a^c = a^b+c, thus you add the powers if they are multiplying and the bases are the same

4 = 2 x 2 = 2¹ x 2¹ = 2¹⁺¹ = 2²

32 = 2 x 2 x 2 x 2 x 2 = 2^1+1+1+1+1 = 2⁵

4^x = 32

⇒ (2²)^x = 2⁵

Applying the law of indices, (aⁿ)^m = a^nxm

⇒ 2^2x = 2⁵

Since the bases are the same, the powers must also be the same

⇒2x = 5

Divide both sides by 2

⇒ x = 5/2 = 2½

Concept

a^b x a^c = a^b+c

a^b ÷ a^c = a^b-c

⇒ 2² x 2⁷ ÷ 2⁴ = 2²⁺⁷ ÷ 2⁴

⇒ 2² x 2⁷ ÷ 2⁴ = 2⁹ ÷ 2⁴

⇒ 2² x 2⁷ ÷ 2⁴ = 2^{9 - 4}

⇒ 2² x 2⁷ ÷ 2⁴ = 2⁵

Difference in mathematics means subtraction.

Product means multiplication.

Sum means addition.

Hence difference between (2d)² and 2d² implies (2d)² - 2d²

NOTE: In (2d)² the square (2) is affecting both 2 and d since they are both in a bracket but in 2d², the square (2) only affects the d.

2d means 2 x d

2d² means 2 times d square, thus 2 x d²

d² = d x d

Hence 2d² = 2 x d x d

In (2d)², you can simplify the 2d and when done square the result, thus multiply the result by itself.

Since d = 3, substitute 3 into wherever you see d and simplify

2d = 2 x d, but d = 3

⇒ 2d = 2 x 3 = 6

(2d)² = (2 x 3)² = 6² = 6 x 6 = 36

2d² = 2 x 3² = 2 x 3 x 3 = 2 x 9 = 18

(2d)² - 2d² = 36 - 18 = 18

Express both sides of equation to have the same base:

Since the least base is 2, express 8 as a power of 2:

8 = 2 x 2 x 2 = 2³

2^2m = 2³

Since the bases are the same, the exponents must be equal:

2m = 3

Divide both sides by 2

m = $\frac{3}{2}$

m = 1.5