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!