## Powers, Roots, and Laws of indices

### Powers, Roots, and Laws of Indices

#### Introduction to Powers and Roots

**Powers (Exponents):**

A power, also known as an exponent, is a way to express repeated multiplication of the same number. For example, \(2^3\) (read as “two to the power of three”) means \(2 \times 2 \times 2 = 8\).

**General Form:**

\[ a^n \]

– \(a\) is the base.

– \(n\) is the exponent.

**Roots**:

A root is the inverse operation of a power. The most common root is the square root. For example, the square root of 9 is 3 because \(3^2 = 9\).

**General Form:**

\[ \sqrt[n]{a} \]

– \(a\) is the radicand.

– \(n\) is the degree of the root (when \(n\) is 2, it’s called the square root; when \(n\) is 3, it’s called the cube root).

#### Laws of Indices

**1. Multiplication Law:**

\[ a^m \times a^n = a^{m+n} \]

When multiplying two powers with the same base, add their exponents.

Example:

\[ 2^3 \times 2^4 = 2^{3+4} = 2^7 \]

**2. Division Law:**

\[ \frac{a^m}{a^n} = a^{m-n} \]

When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.

Example:

\[ \frac{5^6}{5^2} = 5^{6-2} = 5^4 \]

**3. Power of a Power Law:**

\[ (a^m)^n = a^{m \times n} \]

When raising a power to another power, multiply the exponents.

Example:

\[ (3^2)^4 = 3^{2 \times 4} = 3^8 \]

**4. Power of a Product Law:**

\[ (ab)^n = a^n \times b^n \]

When raising a product to a power, raise each factor to the power.

Example:

\[ (2 \times 3)^4 = 2^4 \times 3^4 \]

**5. Power of a Quotient Law:**

\[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

When raising a quotient to a power, raise both the numerator and the denominator to the power.

Example:

\[ \left(\frac{4}{2}\right)^3 = \frac{4^3}{2^3} \]

**6. Zero Exponent Law:**

\[ a^0 = 1 \]

Any non-zero base raised to the power of zero is 1.

Example:

\[ 7^0 = 1 \]

**7. Negative Exponent Law:**

\[ a^{-n} = \frac{1}{a^n} \]

A negative exponent means the reciprocal of the base raised to the opposite positive exponent.

Example:

\[ 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \]

**8. Fractional Exponents:**

\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \]

A fractional exponent represents a root. The numerator is the power, and the denominator is the root.

Example:

\[ 8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4 \]

#### Examples and Practice Problems

1. Simplify \(4^3 \times 4^2\):

\[ 4^3 \times 4^2 = 4^{3+2} = 4^5 = 1024 \]

2. Simplify \(\frac{10^5}{10^2}\):

\[ \frac{10^5}{10^2} = 10^{5-2} = 10^3 = 1000 \]

3. Simplify \((2^3)^2\):

\[ (2^3)^2 = 2^{3 \times 2} = 2^6 = 64 \]

4. Simplify \((3 \times 4)^2\):

\[ (3 \times 4)^2 = 3^2 \times 4^2 = 9 \times 16 = 144 \]

5. Simplify \(\left(\frac{2}{3}\right)^3\):

\[ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} \]

6. Simplify \(5^0\):

\[ 5^0 = 1 \]

7. Simplify \(2^{-3}\):

\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]

8. Simplify \(27^{\frac{1}{3}}\):

\[ 27^{\frac{1}{3}} = \sqrt[3]{27} = 3 \]

Understanding powers, roots, and the laws of indices is essential for simplifying and solving mathematical expressions involving exponents. By applying these laws, complex expressions can be simplified systematically.

### Practice Problems

1. Simplify: \(7^2 \times 7^3\)

2. Simplify: \(\frac{9^4}{9^2}\)

3. Simplify: \((5^2)^3\)

4. Simplify: \((6 \times 2)^2\)

5. Simplify: \(\left(\frac{4}{5}\right)^2\)

6. Simplify: \(3^{-2}\)

7. Simplify: \(16^{\frac{1}{4}}\)

8. Simplify: \(\left(2^3 \times 4^2\right)^2\)

Feel free to ask any questions or request further explanations on any of these topics!