Question

Simplify each expression. $$4^{-3 / 2}$$

Short answer

\( 4^{-3/2} = \frac{1}{8} \)

Step-by-step solution

Understand the Expression

Simplify the given expression \[ 4^{-3/2} \], which is an exponent form with a negative fractional exponent.

Use the Negative Exponent Rule

Rewrite the expression by using the rule that \[ a^{-b} = \frac{1}{a^b} \], so \[ 4^{-3/2} = \frac{1}{4^{3/2}} \].

Use the Fractional Exponent Rule

Next, handle the fractional exponent by recognizing that \[ a^{m/n} = (\sqrt[n]{a})^m \]. Therefore, \[ 4^{3/2} = (\sqrt{4})^3 \].

Simplify the Square Root

Simplify \( \sqrt{4} \), which is \[ 2 \], thus \[ (\sqrt{4})^3 = 2^3 \].

Simplify the Exponent

Simplify \[ 2^3 = 8 \]. So, the original negative exponent expression turns into a fraction \[ \frac{1}{8} \].

Key concepts

negative exponents

Negative exponents might seem tricky at first, but they follow a simple rule that can help you simplify many expressions. When you see an exponent with a negative sign, like in \( 4^{-3/2} \), it means you take the reciprocal of the base raised to the positive version of the exponent. For example, \( a^{-b} = \frac{1}{a^b} \). In our exercise, \( 4^{-3/2} \) becomes \( \frac{1}{4^{3/2}} \). This flip helps make the base positive and easier to work with.

fractional exponents

Fractional exponents are another important topic when simplifying expressions. They represent both powers and roots. For example, \( a^{m/n} \) means taking the \( n \)-th root of the base \( a \) and then raising it to the power of \( m \). In our given problem, \( 4^{3/2} \) translates to \( (\sqrt[2]{4})^3 \). Simplifying the square root first, you get \( \sqrt{4} = 2 \). Now raise this to the power of 3 to get \( 2^3 = 8 \). So, \( 4^{3/2} = 8 \).

radicals

Radicals, such as square roots, are closely related to fractional exponents. They allow you to express roots in a simpler form. The radical symbol \( \sqrt{} \) is commonly used, but you can also express it using fractional exponents. For instance, converting \( \sqrt[2]{4} \) to an exponent notation gives \( 4^{1/2} \). This is very handy when working with more complex expressions. For example, in our problem, \( 4^{3/2} \) can be written as \( (\sqrt[2]{4})^3 \). Knowing how to switch between these forms can make challenging problems much more manageable.

exponent rules

Exponent rules are essential for simplifying expressions efficiently. They include the power rule, product rule, quotient rule, and of course, handling negative and fractional exponents. The power rule, which states \( (a^m)^n = a^{m*n} \), is particularly useful when dealing with stacked exponents. The product rule \( a^m \times a^n = a^{m+n} \) and quotient rule \( \frac{a^m}{a^n} = a^{m-n} \) help you combine or divide like bases. Understanding these rules lets you break down even the most complex expressions systematically.