Question
How do you simplify expressions with negative exponents in algebra?
Answers
GPT-4o mini
To simplify expressions with negative exponents in algebra, you can use the following rules:
1. **Understanding Negative Exponents**: A negative exponent indicates that you take the reciprocal of the base and then raise it to the corresponding positive exponent. Thus, for any non-zero number \( a \) and any integer \( n \):
\[
a^{-n} = \frac{1}{a^n}
\]
2. **Applying the Rule**: If you encounter a negative exponent in an expression, you can rewrite it using the formula above. For example:
- If you have \( x^{-3} \), it can be rewritten as \( \frac{1}{x^3} \).
- For more complex expressions, such as \( \frac{2}{x^{-2}} \), you would rewrite it as \( \frac{2}{\frac{1}{x^2}} = 2 \cdot x^2 = 2x^2 \).
3. **Combining Terms**: When simplifying expressions with both positive and negative exponents, use the rules of exponents. For example:
- If you have \( x^{2} \cdot x^{-3} \), you can combine them by subtracting the exponents:
\[
x^{2} \cdot x^{-3} = x^{2 + (-3)} = x^{-1} = \frac{1}{x}
\]
4. **Multiple Variables**: If the expression contains multiple variables, apply the same rules. For example:
- Consider \( x^{-2}y^{3} \). This can be expressed as:
\[
\frac{y^{3}}{x^{2}}
\]
5. **Fractional Expressions**: If negative exponents appear in the denominator, follow the same rule. For instance, if you have:
- \( \frac{1}{x^{-3}} \), you can rewrite that as:
\[
x^{3}
\]
because \( x^{-3} \) in the denominator becomes positive in the numerator.
6. **Combining Everything**: When dealing with a more complex expression like:
\[
\frac{x^{-1}y^2}{z^{-3}}
\]
you would first rewrite it with positive exponents:
\[
\frac{y^2}{x} \cdot z^3 = \frac{y^2 z^3}{x}
\]
By following these steps, you can systematically simplify expressions with negative exponents in algebra.
1. **Understanding Negative Exponents**: A negative exponent indicates that you take the reciprocal of the base and then raise it to the corresponding positive exponent. Thus, for any non-zero number \( a \) and any integer \( n \):
\[
a^{-n} = \frac{1}{a^n}
\]
2. **Applying the Rule**: If you encounter a negative exponent in an expression, you can rewrite it using the formula above. For example:
- If you have \( x^{-3} \), it can be rewritten as \( \frac{1}{x^3} \).
- For more complex expressions, such as \( \frac{2}{x^{-2}} \), you would rewrite it as \( \frac{2}{\frac{1}{x^2}} = 2 \cdot x^2 = 2x^2 \).
3. **Combining Terms**: When simplifying expressions with both positive and negative exponents, use the rules of exponents. For example:
- If you have \( x^{2} \cdot x^{-3} \), you can combine them by subtracting the exponents:
\[
x^{2} \cdot x^{-3} = x^{2 + (-3)} = x^{-1} = \frac{1}{x}
\]
4. **Multiple Variables**: If the expression contains multiple variables, apply the same rules. For example:
- Consider \( x^{-2}y^{3} \). This can be expressed as:
\[
\frac{y^{3}}{x^{2}}
\]
5. **Fractional Expressions**: If negative exponents appear in the denominator, follow the same rule. For instance, if you have:
- \( \frac{1}{x^{-3}} \), you can rewrite that as:
\[
x^{3}
\]
because \( x^{-3} \) in the denominator becomes positive in the numerator.
6. **Combining Everything**: When dealing with a more complex expression like:
\[
\frac{x^{-1}y^2}{z^{-3}}
\]
you would first rewrite it with positive exponents:
\[
\frac{y^2}{x} \cdot z^3 = \frac{y^2 z^3}{x}
\]
By following these steps, you can systematically simplify expressions with negative exponents in algebra.