Question

How can you simplify and evaluate expressions with zerp and negative exponents?

Answer 1

summerize this please

Answer 2

To simplify and evaluate expressions with zero and negative exponents, you can follow these general rules:

### 1. Zero Exponent Rule:
Any non-zero number raised to the power of zero is equal to one.
\[
a^0 = 1 \quad (\text{for } a \neq 0)
\]

### 2. Negative Exponent Rule:
A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.
\[
a^{-n} = \frac{1}{a^n} \quad (\text{for } a \neq 0)
\]

### Steps to Simplify and Evaluate Expressions:
1. **Identify the Exponents**: Look for zero and negative exponents in the expression.
2. **Apply the Zero Exponent Rule**: Replace any terms with a zero exponent with 1.
3. **Apply the Negative Exponent Rule**: Convert any terms with negative exponents into their reciprocal form.
4. **Simplify Further if Needed**: Combine like terms or simplify fractions as necessary.

### Example 1:
Simplify \( 3^2 \cdot 3^0 \cdot 3^{-3} \).

**Step 1**: Identify exponents:
- \( 3^2 \): 2 is a positive exponent.
- \( 3^0 \): 0 exponent.
- \( 3^{-3} \): Negative exponent.

**Step 2**: Apply the Zero Exponent Rule:
- \( 3^0 = 1 \)

**Step 3**: Apply the Negative Exponent Rule:
- \( 3^{-3} = \frac{1}{3^3} \)

**Rewrite the expression**:
\[
3^2 \cdot 1 \cdot \frac{1}{3^3}
\]

**Step 4**: Simplify further:
\[
= \frac{3^2}{3^3} = 3^{2-3} = 3^{-1} = \frac{1}{3}
\]

### Example 2:
Evaluate \( 5^{-2} + 4 \cdot 5^0 \).

**Step 1**: Identify exponents:
- \( 5^{-2} \): Negative exponent.
- \( 5^0 = 1 \).

**Step 2**: Apply the Negative Exponent Rule:
- \( 5^{-2} = \frac{1}{5^2} = \frac{1}{25} \)

**Step 3**: Calculate \( 4 \cdot 5^0 \):
- \( 4 \cdot 5^0 = 4 \cdot 1 = 4 \)

**Rewrite the total expression**:
\[
\frac{1}{25} + 4
\]

**Convert 4 to a fraction with a common denominator**:
\[
4 = \frac{100}{25}
\]

**Final calculation**:
\[
\frac{1}{25} + \frac{100}{25} = \frac{101}{25}
\]

### Conclusion:
By using the zero and negative exponent rules, you can simplify and evaluate expressions systematically. Always ensure the base is not zero when applying these rules.

Answer 3

To simplify and evaluate expressions with zero and negative exponents, follow these key rules:

### Key Rules:
1. **Zero Exponent Rule**: Any non-zero number raised to the power of zero is 1:
\[
a^0 = 1 \quad (\text{for } a \neq 0)
\]

2. **Negative Exponent Rule**: A negative exponent represents the reciprocal of the base raised to the positive exponent:
\[
a^{-n} = \frac{1}{a^n} \quad (\text{for } a \neq 0)
\]

### Steps for Simplification:
1. **Identify Exponents**: Find zero and negative exponents in the expression.
2. **Apply Rules**: Replace terms with \(0\) exponents with \(1\) and convert negative exponents to their reciprocal forms.
3. **Simplify Further**: Combine like terms or simplify any fractions as needed.

### Examples:
- For \( 3^2 \cdot 3^0 \cdot 3^{-3} \):
- Simplifies to \( \frac{1}{3} \).

- For \( 5^{-2} + 4 \cdot 5^0 \):
- Simplifies to \( \frac{101}{25} \).

By using these steps and rules, you can effectively simplify and evaluate expressions with zero and negative exponents.