Question

\(\left(5^{1 / 2} \cdot 5^{-3 / 2}\right)^{-1 / 4}\)

Short answer

The simplified form of \(\left(5^{1 / 2} \cdot 5^{-3 / 2}\right)^{-1 / 4}\) is \(5^{1 / 4}\).

Step-by-step solution

Combine Exponents by Addition and Subtraction

First combine the \(5^{1 / 2}\) and \(5^{-3 / 2}\) by subtracting the fraction exponents because the bases are being multiplied together. This gives us: \(5^{1 / 2 - 3 / 2} = 5^{-1}\).

Apply Power of a Power Rule

Following this, apply the power of a power rule to \(5^{-1}^{-1 / 4}\), which means multiplying the exponents together. This gives us: \(5^{-1*-1 / 4} = 5^{1 / 4}\).

Simplify

Finally, simplify \(5^{1 / 4}\), as nothing else adheres to the rules of exponents within this expression.

Key concepts

Power of a Power Rule

When we encounter expressions where an exponent is raised to another exponent, like \( b^{m}^{n} \), the power of a power rule comes into play. In these cases, instead of working out one exponent and then the next, the rule states that we can simply multiply the exponents together: \( b^{m}^{n} = b^{m \cdot n} \).

This rule helps to greatly simplify calculations and is especially useful when dealing with fractional exponents or negative exponents. For instance, if you have an expression like \( (5^{1/2})^{-1/4} \), we multiply the exponents \( 1/2 \) and \( -1/4 \), which simplifies to \( 5^{1/2 \cdot -1/4} = 5^{-1/8} \). This step is fundamental in simplifying exponential expressions and is inseparable from the process of working through more complex algebraic problems.

Combining Exponents

When you have the same base and are either multiplying or dividing the expressions, you can combine exponents by addition or subtraction respectively. If the bases are being multiplied, you add the exponents: \( b^{m} \cdot b^{n} = b^{m+n} \). Conversely, if the bases are being divided, you subtract the exponents: \( \frac{b^{m}}{b^{n}} = b^{m-n} \).

This concept is crucial when simplifying expressions with multiple exponential terms. For example, in our exercise, when we combine \( 5^{1/2} \) and \( 5^{-3/2} \) we are effectively dealing with multiplication, hence we add the exponents: \( 5^{1/2 + (-3/2)} \), which simplifies to \( 5^{-1} \). It is important to pay close attention to the signs of the exponents during this process as they significantly impact the final result.

Simplifying Exponential Expressions

The key to simplifying exponential expressions lies in applying a series of exponent rules methodically. After combining exponents or using the power of a power rule, the expression should be simplified to its most basic form. This often means converting negative exponents to positive by taking the reciprocal of the base and eliminating any fractional exponents by finding the corresponding root of the base, if possible.

In the case of our exercise, once we've applied the previous rules, we are left with \( 5^{1/4} \), which is already in its simplest form since it represents the fourth root of 5, a number that cannot be simplified further without a calculator. It’s important to remember that not all expressions will simplify to an integer or a simple fraction, and in such cases, it's perfectly acceptable to leave the answer in exponential form with a positive exponent, if applicable.