Subject: Compulsory Maths

The laws of indices state a number of rules, which can be used to simplify expressions involving indices. Here, an index is used to write a product of numbers very compactly. The plural of index is indices.

Indices is a number with the power. For example: a^{m}; a is called the base and m is the power. These laws only apply to expression with the same base.

Index help to write a product of numbers very compactly. Index help to show how many times to use the number in a multiplication. It is shown in the top right of the number in small number.

In this example: 4³ = 4x4x4 = 64

Any number, except 0, whose index is 0 is always equal to 1.

**An example:**

2° = 1

**An example:**

2^{-3 }= \(\frac{1}{2^3}\) ( using a^{-m }= \(\frac{1}{a^m}\))

In case of multiplication of same base, copy the base and add the indices.

**An example:**

3^{2 }x 3^{4} = 3^{2+4 }(using a^{m} x a^{n }= a^{ m+n})

= 3^{6}

= 3 x 3 x 3 x 3 x 3 x 3

= 729

In case of division of same base, copy the base and subtract the indices.

**An example:**

w^{10 }÷ w^{6}= w^{10-6} = w^{4}

To raise an expression to the n^{th} index, Copy the base and multiply the indices.

**An example:**

( x^{2})^{4} = x^{2x4 }= x^{8}

**An example:**

125^{\(\frac{2}{3}\) }= (\(\sqrt[3]{125}\))^{2} = (5)^{2} = 25

- An indices is a number with the power.
- The laws of indices state a number of rules, which can be used to simplify expressions involving indices.
- Any number, except 0, whose index is 0 is always equal to 1. (i.e. a° = 1)

- It includes every relationship which established among the people.
- There can be more than one community in a society. Community smaller than society.
- It is a network of social relationships which cannot see or touched.
- common interests and common objectives are not necessary for society.

Solve by using laws of indices.

8^{4}×8^{3}

Solution:

8^{4}×8^{3}

=8^{4+3}

=8^{7}

Solve by using laws of indices.

a^{6}×a^{7}

Solution,

a^{6}×a^{7}

=a^{6+7}

=x^{13}

Solve by using laws of indices.

(x^{3}y)×(xy)×(x^{2}y)

Solution,

(x^{3}y)×(xy)×(x^{2}y)

= x^{3}× x× x^{2}×y×y×y

=x^{3+1+2}×b^{1+1+1}

= x^{6}y^{3}

Solve by using laws of indices.

3^{5 }÷3^{3}

Solution:

3^{5}÷3^{3}

=\(\frac{3^5}{3^3}\)

=3^{5-3}

=3^{2}

=9

Solve by using laws of indices.

12x^{7}÷3x^{5}

Solution:

12x7÷3x^{5}

=\(\frac{12x^7}{3x^5}\)

=\(\frac{3×4x^7-5}{3}\)

=4x^{2}

Solve by using laws of indices.

\(\frac{a^{m+n+2}×a^{m+n+2}}{a^{m+n}}\)

Solution:

\(\frac{a^{m+n+2}×a^{m+n+2}}{a^{m+n}}\)

=\(\frac{a(^{(m+n+2)} \times\ a^{(m+n+2)}}{a^{m+n}}\)

=a^{2(m+n+2)-(m+n)}=a^{2m+2n+4-m-n}

=a^{m+n+4}

Solve by using laws of indices.

-36a^{8}÷9a^{5}

Solution:

\(\frac{-36a^8}{9a^5}\)

=\(\frac{-4×9a^{8-5}}{9}\)

= -4a^{3}

Solve by using laws of indices.

(2^{3})^{4}

Solution:

(2^{3})^{4}

=2^{3×4}

=2^{12}

Solve by using laws of indices.

(5x^{3})^{4}

Solution:

(5x^{3})^{4}

=5^{4}x^{3×}^{4}

=5^{4}x^{12}

Solve by using laws of indices.

(a^{2}b)^{c}×(ab^{2})^{c}

Solution:

(a^{2}b)^{c}×(ab^{2})^{c}

=(a^{2})^{c}.b^{c}×a^{c}(b^{2})^{c}

=a^{2c}.b^{c}×a^{c}.b^{2c}

=a^{2c+c}.b^{c+2c}

=a^{3c}.b^{3c}

Find the value by using the law of indices.

\(\frac{2^3×4^2}{8^2}\)

Solution:

\(\frac{2^3×4^2}{8^2}\)

=\(\frac{2^3.(2^2)^2}{(2^3)^2}\)

=\(\frac{2^3.2^4}{2^6}\)

=2^{3+4-6}

=2^{7-6}

=2^{1}

=2

Solve by using the law of indices.

(a^{2}b)×(ab)

Solution:

(a^{2}b)×(ab)

=a^{2}×a×b×b

=a^{2+1}×b^{1+1}

=a^{3}b^{2}

Find the value by using the law of indices.

\(\frac{5^3×125^3}{25^3}\)

Solution:

\(\frac{5^3×125^3}{25^3}\)

=\(\frac{5^3×(5^3)^3}{(5^2)^3}\)

=\(\frac{5^3×5^9}{5^6}\)

=5^{3+9-6}

=5^{12-6}

=5^{6}

Find the value by using the law of indices.

\(\frac{4^4×5^5}{25^2×16^2}\)

Solution:

\(\frac{4^4×5^5}{25^2×16^2}\)

=\(\frac{4^4×5^5}{(5^2)^2×(4^2)^2}\)

=\(\frac{4^4×5^5}{5^4×4^4}\)

=4^{4-4}×5^{5-4}

=4^{0}×5^{1}

=1×5

=5

Solveby using the law of indices.

(-7p^{3})^{4}

Solution:

(-7p^{3})^{4}

=(-7)^{4}.(p^{3})^{4}

=7^{4}.p^{3×4}

=7^{4}p^{12}

Solve by using the law of indices.

(xy^{2})^{3}×xy

Solution:

(xy^{2})^{3}×xy

=x^{3}(y^{2})^{3}×xy

=x^{3}y^{2×3}×xy

=x^{3}.x.y^{6}.y

=x^{3+1}.y^{6+1}

=x^{4}y^{7}

Solve by using the law of indices.

(4x^{4})×(3x^{3})^{4}

Solution:

(4x^{4})×(3x^{3})^{4}=4^{3}(x^{4})^{3}×3^{4}(x^{3})4

=4^{3}.x^{4×3}.3^{4}.x^{3×4}

=64×81.x^{12}.x^{12}

=4^{3}×3^{4}x^{12+12}

=4^{3}×3^{4} x^{24}

Solve the following by using the law of indices.

\(\frac{(3p^2q)^2}{p^2q^2}\)

Solution:

\(\frac{(3p^2q)^2}{p^2q^2}\)

\(\frac{3^2(p^2)^2.q^2}{9p^2q^2}\)

=\(\frac{9p^4.q^2}{9p^2q^2}\)

=p^{4-2}q^{2-2}

=p^{2}.q^{0}

=p^{2}

Solve by using laws of indices.

a^{5}×a^{10}

Solution,

a^{5}×a^{10}

=a^{5+10}

=a^{15}

Solve by using laws of indices.

z^{4} x z^{2}

Solution,

z^{4} x z^{2}

= z^{4+}^{2}

=z^{6}

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