15 Aug, 2024

· Mathematics

What expression is equivalent to 4 to the negative 3 power

Short Answer

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Long Explanation

Explanation

Equivalent Expression

To find the expression equivalent to $4^{-3}$ :

We use the property of exponents which states:

a^{-n} = \frac{1}{a^n}

Applying this property:

4^{-3} = \frac{1}{4^3}

Calculating $4^3$ :

4^3 = 4 \times 4 \times 4 = 64

Thus:

4^{-3} = \frac{1}{64}

So, the expression equivalent to $4^{-3}$ is $\frac{1}{64}$ .

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Emily Rosen

Mathematics Content Writer at Math AI

Emily Rosen is a recent graduate with a Master's in Mathematics from the University of Otago. She has been tutoring math students and working as a part-time contract writer for the past three years. She is passionate about helping students overcome their fear of math through easily digestible content.

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Concept

Exponents And Negative Exponents

Rule for Exponents and Negative Exponents

Exponents are a shorthand way to represent repeated multiplication of a number by itself. For a base $a$ raised to the power of $n$ , we write $a^n$ , which means:

a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}

Positive Exponents

For positive exponents, $a^3$ means multiplying the base $a$ three times:

a^3 = a \times a \times a

Zero Exponent

Any non-zero base raised to the power of zero is always 1:

a^0 = 1 \quad \text{for } a \neq 0

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. For a base $a$ raised to the power of $-n$ , we write $a^{-n}$ , which means:

a^{-n} = \frac{1}{a^n}

For example:

a^{-2} = \frac{1}{a^2}

Example Calculation

Let’s calculate some examples to illustrate these rules:

$2^3 = 2 \times 2 \times 2 = 8$
$5^0 = 1$
$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

Combining Exponents

When multiplying or dividing like bases, you can combine the exponents:

Multiplication: $a^m \times a^n = a^{m+n}$

2^3 \times 2^2 = 2^{3+2} = 2^5 = 32

Division: $\frac{a^m}{a^n} = a^{m-n}$

\frac{3^4}{3^2} = 3^{4-2} = 3^2 = 9

Understanding these rules allows you to simplify and compute expressions involving exponents and negative exponents efficiently.

Concept

Property Of Exponents

Explanation

The distributive property of exponents allows you to simplify expressions where exponents are distributed across multiplication or division inside parentheses. Here’s a detailed look at this property:

Distributive Property Over Multiplication

For any non-zero numbers $a$ and $b$ , and a rational number $n$ :

(a \cdot b)^n = a^n \cdot b^n

Example:

(2 \cdot 3)^4 = 2^4 \cdot 3^4 = 16 \cdot 81 = 1296

Distributive Property Over Division

For any non-zero numbers $a$ and $b$ , and a rational number $n$ :

\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Example:

\left(\frac{4}{2}\right)^3 = \frac{4^3}{2^3} = \frac{64}{8} = 8

Key Points

The property only works correctly if the base numbers are non-zero.
Negative exponents and fractional exponents still adhere to this property, though you must be cautious in their application.

This property greatly aids in simplifying complex exponential expressions and solving exponential equations.