Simplify the following expression. $$ 9^{-1}+4^{-1} $$

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UpStudy AI · Accepted Answer

To simplify the expression:

$$
9^{-1} + 4^{-1}
$$

let's break it down step by step.

### Step 1: Understand Negative Exponents
A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. Therefore:

$$
9^{-1} = \frac{1}{9}
$$
$$
4^{-1} = \frac{1}{4}
$$

### Step 2: Rewrite the Expression with Fractions
Substitute the negative exponents with their fractional equivalents:

$$
9^{-1} + 4^{-1} = \frac{1}{9} + \frac{1}{4}
$$

### Step 3: Find a Common Denominator
To add the two fractions, they must have the same denominator. The least common denominator (LCD) of 9 and 4 is 36.

$$
\frac{1}{9} = \frac{4}{36} \quad \text{(multiplied numerator and denominator by 4)}
$$
$$
\frac{1}{4} = \frac{9}{36} \quad \text{(multiplied numerator and denominator by 9)}
$$

### Step 4: Add the Fractions
Now that both fractions have the same denominator, add them together:

$$
\frac{4}{36} + \frac{9}{36} = \frac{4 + 9}{36} = \frac{13}{36}
$$

### Final Simplified Expression
$$
9^{-1} + 4^{-1} = \frac{13}{36}
$$

**Answer:** $$\frac{13}{36}$$
$$ 9^{-1} + 4^{-1} = \frac{13}{36} $$

Simplify the following expression.

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