Question
What is laws of exponents in math
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Answered by
GPT-4o mini
The laws of exponents, also known as the rules of exponents, are fundamental properties that describe how to handle mathematical expressions involving exponents or powers. Here are the main laws of exponents:
1. **Product of Powers**:
\[
a^m \times a^n = a^{m+n}
\]
When multiplying two powers with the same base, you add the exponents.
2. **Quotient of Powers**:
\[
\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)
\]
When dividing two powers with the same base, you subtract the exponents.
3. **Power of a Power**:
\[
(a^m)^n = a^{m \times n}
\]
When raising a power to another power, you multiply the exponents.
4. **Power of a Product**:
\[
(ab)^n = a^n \times b^n
\]
When raising a product to a power, you distribute the exponent to each factor.
5. **Power of a Quotient**:
\[
\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \quad (b \neq 0)
\]
When raising a quotient to a power, you distribute the exponent to the numerator and denominator.
6. **Zero Exponent**:
\[
a^0 = 1 \quad (a \neq 0)
\]
Any non-zero number raised to the power of zero is 1.
7. **Negative Exponent**:
\[
a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
\]
A negative exponent represents the reciprocal of the base raised to the opposite positive exponent.
These rules allow you to simplify expressions involving exponents and perform calculations more easily.
1. **Product of Powers**:
\[
a^m \times a^n = a^{m+n}
\]
When multiplying two powers with the same base, you add the exponents.
2. **Quotient of Powers**:
\[
\frac{a^m}{a^n} = a^{m-n} \quad (a \neq 0)
\]
When dividing two powers with the same base, you subtract the exponents.
3. **Power of a Power**:
\[
(a^m)^n = a^{m \times n}
\]
When raising a power to another power, you multiply the exponents.
4. **Power of a Product**:
\[
(ab)^n = a^n \times b^n
\]
When raising a product to a power, you distribute the exponent to each factor.
5. **Power of a Quotient**:
\[
\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \quad (b \neq 0)
\]
When raising a quotient to a power, you distribute the exponent to the numerator and denominator.
6. **Zero Exponent**:
\[
a^0 = 1 \quad (a \neq 0)
\]
Any non-zero number raised to the power of zero is 1.
7. **Negative Exponent**:
\[
a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
\]
A negative exponent represents the reciprocal of the base raised to the opposite positive exponent.
These rules allow you to simplify expressions involving exponents and perform calculations more easily.