Question

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

Short answer

The simplified expression is 8.

Step-by-step solution

Understand the Expression

The given expression is in the form of a power, specifically a fractional exponent. The expression to simplify is \(4^{3 / 2}\). The general form of a fractional exponent \(a^{m/n}\) can be interpreted as taking the \(n\)-th root of \(a\) and then raising the result to the \(m\)-th power.

Rewrite the Fractional Exponent

Rewrite the expression \(4^{3 / 2}\) in terms of roots and powers. According to the rules of exponents, \(4^{3 / 2}\) is equivalent to \(\big(4^{1/2}\big)^3\). This means we take the square root of 4, then raise the result to the power of 3.

Find the Square Root

Calculate the square root of 4. \(\sqrt{4} = 2\). So now the expression becomes \(2^3\).

Raise to the Power of 3

Raise 2 to the power of 3. \(2^3 = 8\).

State the Simplified Expression

Thus, the simplified expression of \(4^{3 / 2}\) is \(8\).

Key concepts

fractional exponents

Fractional exponents might look complicated, but they are a useful way to represent roots and powers in one compact form.
When you see an exponent written as a fraction, such as \(a^{m/n}\), it can be broken down into two parts:

• The numerator (m) which indicates the power
• The denominator (n) which indicates the root

For example, \(4^{3/2}\) can be read as taking the square root of 4 and then raising that result to the 3rd power.
This means that: \[ 4^{3/2} = \big(4^{1/2}\big)^3 \]
Understanding this basic principle can make simplifying expressions with fractional exponents much easier. It's all about breaking them down into understandable parts and then solving step by step. Remember, practice makes perfect!

square root

The square root is the value that, when multiplied by itself, gives the original number.
It is denoted by the radical symbol, \( \sqrt{} \), where \( \sqrt{4} = 2 \) because \(2 \times 2 = 4 \).
In our expression, \(4^{3/2}\), taking the square root is the first step:

• This means we need to find the number which when squared equals 4.
• The square root of 4 is \(\sqrt{4} = 2\), since \(2 \times 2 = 4\).

So, \(4^{1/2} = 2\).
It’s important to understand this operation, as it comes up frequently when dealing with both fractional exponents and even more complex algebraic expressions. Simplifying the square root allows us to move on to the next steps in solving.

power rules

Power rules help us understand how to manipulate exponents efficiently.
In our exercise, after finding the square root, we need to raise the result to a power: \[ (\sqrt{4})^3 = 2^3 \]
Here are some important power rules:

• \(a^{m \times n} = (a^m)^n\): This tells us how to handle nested exponents.
• \((ab)^m = a^m b^m\): When raising a product to a power, you raise each factor to the power.
• \(a^{-m} = 1/a^m\): Negative exponents indicate reciprocal.

Applying the first rule, we continue simplifying: \[ 2^3 = 2 \times 2 \times 2 = 8 \]
The final expression is \(8\). Understanding power rules not only helps solve problems like these but builds a foundation for more advanced mathematics.