Question
if the bases are the same with exponenets
Answers
GPT-4o mini
When the bases are the same in an expression involving exponents, you can apply the laws of exponents to simplify the expression. Here are some key rules to remember:
1. **Multiplication of like bases**: When you multiply two expressions with the same base, you add the exponents.
\[
a^m \cdot a^n = a^{m+n}
\]
2. **Division of like bases**: When you divide two expressions with the same base, you subtract the exponents.
\[
\frac{a^m}{a^n} = a^{m-n}
\]
3. **Power of a power**: When you raise an exponent to another exponent, you multiply the exponents.
\[
(a^m)^n = a^{m \cdot n}
\]
4. **Power of a product**: When you have a product raised to an exponent, you can distribute the exponent to each factor inside the parentheses.
\[
(ab)^n = a^n \cdot b^n
\]
5. **Power of a quotient**: When you have a quotient raised to an exponent, you can apply the exponent to both the numerator and the denominator.
\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
\]
Using these properties allows you to simplify expressions involving exponents when the bases are the same.
1. **Multiplication of like bases**: When you multiply two expressions with the same base, you add the exponents.
\[
a^m \cdot a^n = a^{m+n}
\]
2. **Division of like bases**: When you divide two expressions with the same base, you subtract the exponents.
\[
\frac{a^m}{a^n} = a^{m-n}
\]
3. **Power of a power**: When you raise an exponent to another exponent, you multiply the exponents.
\[
(a^m)^n = a^{m \cdot n}
\]
4. **Power of a product**: When you have a product raised to an exponent, you can distribute the exponent to each factor inside the parentheses.
\[
(ab)^n = a^n \cdot b^n
\]
5. **Power of a quotient**: When you have a quotient raised to an exponent, you can apply the exponent to both the numerator and the denominator.
\[
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
\]
Using these properties allows you to simplify expressions involving exponents when the bases are the same.