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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