MATHCRAVE ALGEBRA
Power, Roots and Indices
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# Powers, Roots, and Laws of indices

## 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!