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# Indices

In mathematics, indices provide a convenient way to denote repeated multiplication. Commonly referred to as powers or exponents, they become pivotal when working in algebra, calculus, and various other domains of math.

## Basic Rules of Indice

1. Multiplication with the same base:

When multiplying two terms with the same base, the base remains unchanged, and their powers (indices) are added together.

Rule:

$$a^m \times a^n = a^{m+n}$$

Example 1:
$$3^2 \times 3^3 = 3^{2+3} = 3^5 = 243$$

Example 2:
$$x^4 \times x^5 = x^{4+5} = x^9$$

2. Division with the same base:

When dividing two terms with the same base, the base remains unchanged, and the exponent of the denominator is subtracted from that of the numerator.

Rule:

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

Example 1:
$$\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25$$

Example 2:

$$\frac{y^7}{y^3} = y^{7-3} = y^4$$

3. Power of a power:
When an exponential expression (a number with an exponent) is raised to another power, you multiply the exponents together.

Rule:

If $$(a^m)^n$$, then it can be written as $$a^{m \times n}$$

Explanation:
Raising an exponent to another power means you’re multiplying the base by itself multiple times.

Example 1:

$$(2^2)^3 = 2^{2 \times 3} = 2^6$$

This results in multiplying 2 by itself six times, equaling 64.

Example 2:

$$(x^3)^4 = x^{3 \times 4} = x^{12}$$

This represents $$x$$ multiplied by itself twelve times.

4. Negative powers:
Any term with a negative exponent can be represented as the reciprocal of the term with the corresponding positive exponent.

Formula: $$a^{-n} = \frac{1}{a^n}$$
Example 1: $$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$

Example 2: $$m^{-3} = \frac{1}{m^3}$$

### Activities:

1. Evaluate and Simplify: $$4^3 \times 4^4$$

Hint: When multiplying two numbers with the same base, add their exponents.

$$a^m \times a^n = a^{m+n}$$

E.g. $$2^3 \times 2^2 = 2^{3+2} = 2^5$$

E.g. $$x^4 \times x^3 = x^{4+3} = x^7$$

2. Evaluate and Simplify: $$\frac{6^5}{6^2}$$

Hint: For division with the same base, subtract the exponent of the denominator from the exponent of the numerator.

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

E.g. $$\frac{5^4}{5^2} = 5^{4-2} = 5^2$$

E.g. $$\frac{y^6}{y^4} = y^{6-4} = y^2$$

3. Evaluate and Simplify:

$$(3^2)^3$$

Hint: When a power is raised to another power, multiply the exponents.

$$(a^m)^n = a^{m \times n}$$

E.g. $$(x^2)^3 = 2^{2 \times 3} = 2^6$$

E.g. $$(a^3)^4 = a^{3 \times 4} = a^{12}$$

4. Evaluate and Simplify:

$$y^{-4} \times y^6$$

Hint: When multiplying two numbers with the same base, add their exponents.

$$a^m \times a^n = a^{m+n}$$

E.g. $$3^{-2} \times 3^5 = 3^{-2+5} = 3^3$$

E.g. $$m^{-1} \times m^3 = m^{-1+3} = m^2$$

5. Evaluate and Simplify:

$$(n^3)^2$$

Hint: When a power is raised to another power, multiply the exponents.

$$(a^m)^n = a^{m \times n}$$

E.g. $$(4^2)^2 = 4^{2 \times 2} = 4^4$$

E.g. $$(z^5)^2 = z^{5 \times 2} = z^(10)$$

### Tricky Exam-Style Questions:

1. If $$2^{x+1} = 32$$, determine x.

Rule Applied: In this case, we need to express 32 as a power of 2, then compare the exponents.

More on this: $$2^{y+2} = 16$$, determine y.

Another one: $$a^{n+3} = a^8$$, determine n.

2. Simplify: $$\frac{a^3 \times a^4}{a^5}$$.

Rule Applied: First, combine the powers using multiplication rule and then use the division rule.

More on this: Simplify $$\frac{3^2 \times 3^3}{3^4}$$

Another one: Simplify $$\frac{b^4 \times b^5}{b^8}$$

3. Given $$b^{m-2} \times b^{3m+1} = b^{17}$$, find the value of m.

Rule Applied: Combine the exponents using the multiplication rule and equate to given exponent.

More on this: Given that $$2^{x-3} \times 2^{4x+2} = 2^{18}$$, find x.

Another one: Given that $$c^{n-1} \times c^{2n+2} = c^{14}$$, find n.

















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