Which expression is equivalent to 6^-3

Equivalent Expression to $6^{-3}$

To find an expression that is equivalent to $6^{-3}$, we need to understand the rules of exponents, particularly negative exponents.

A negative exponent indicates that the base should be taken as the reciprocal. Therefore:

Now, we can further simplify this expression by calculating $6^3$.

Thus, the expression for $6^{-3}$ can be rewritten as:

Where:

• $6^{-3}$ is the original expression with a negative exponent.
• $\frac{1}{6^3}$ represents the reciprocal of $6^3$.
• $6^3$ equals $216$.

Therefore, the expression equivalent to $6^{-3}$ is:

Negative Exponents

Negative exponents indicate that the base number should be divided, rather than multiplied. In simpler terms, a negative exponent represents the reciprocal of the base raised to the corresponding positive exponent. This is a fundamental concept in algebra and is expressed mathematically as:

where $a$ is any non-zero number and $n$ is a positive integer.

Important Formula

When dealing with negative exponents, keep in mind the following general formula:

Examples for Clarity

1. $5^{-2}$:

   $$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$

2. $2^{-3}$:

   $$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$

Key Points

• Reciprocal Relationship: A negative exponent translates to the reciprocal of the base to a positive exponent.

• Simplification: It often helps in simplifying complex expressions, especially in equations involving multiple negative exponents.

By understanding these principles, you can manipulate and simplify expressions involving negative exponents with confidence.

Explanation

In mathematics, a reciprocal is essentially the multiplicative inverse of a number. This means that when you multiply a number by its reciprocal, the result is 1. If you have a non-zero number $a$, its reciprocal is denoted as $\frac{1}{a}$.

Important Formula

The fundamental property of the reciprocal can be expressed as:

Reciprocal of Fractions

For a fraction $\frac{a}{b}$:

Reciprocal of Negative Numbers

For any non-zero negative number $-a$:

Special Note on Zero

The number $0$ does not have a reciprocal because division by zero is undefined.

Examples

1. The reciprocal of $5$ is $\frac{1}{5}$.
2. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.
3. The reciprocal of $-2$ is $-\frac{1}{2}$.