Problem 123 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ \frac{1}{4^{-5 / 2}} $$

Step-by-Step Solution

Answer

The expression simplifies to 32.

Step 1: Understand the Negative Exponent

A negative exponent indicates that you take the reciprocal of the base. Therefore, for any base \(a\) and negative exponent \(-n\), the expression can be rewritten as \(a^{-n} = \frac{1}{a^n}\). So, for \(4^{-5/2}\), converting the negative exponent gives us \(\frac{1}{4^{5/2}}\). This means \(\frac{1}{4^{-5/2}} = 4^{5/2}\).

Step 2: Convert to Radical Form

The expression \(4^{5/2}\) can be rewritten in radical form. A fractional exponent \(n/m\) means taking the \(m\)-th root of the base raised to the \(n\)-th power. So \(4^{5/2}\) is equivalent to \((\sqrt{4})^5\).

Step 3: Simplify the Square Root

Calculate \(\sqrt{4}\), which is the square root of 4. Since 4 is a perfect square, \(\sqrt{4} = 2\). Then, substitute back to get \((\sqrt{4})^5 = 2^5\).

Step 4: Calculate the Final Power

Raise 2 to the 5th power to finish the simplification: \(2^5 = 32\).

Key Concepts

Negative Exponents

Negative exponents can seem tricky at first, but they're simpler than they look. When you see a negative exponent like \(-n\), it signifies the reciprocal of the base raised to the corresponding positive exponent. This means that \(a^{-n} = \frac{1}{a^n}\). A negative exponent flips the base to the bottom of a fraction if it isn't already there.

• This flip changes the position from a numerator to a denominator (or vice-versa) without changing the original base value but switching the sign of its exponent.
• Understanding this mechanism helps simplify many expressions effectively and aids in converting them into a more manageable form.

For example, converting \(4^{-5/2}\) to its reciprocal form gives us \(\frac{1}{4^{5/2}}\). Knowing how to handle negative exponents is crucial for simplifying more complex expressions and is a foundational skill in algebra.

Fractional Exponents

Fractional exponents indicate that an operation involves both a power and a root. This dual nature allows us to express radical expressions in an alternate form. When you see an exponent written as a fraction, like \(n/m\), it translates to taking the \(m\)-th root of the base raised to the \(n\)-th power:

• The numerator (in this case \(n\)) tells you the power to which the base is raised.
• The denominator (such as \(m\)) tells you the root that needs to be taken.

For example, \(4^{5/2}\) can be interpreted as \((\sqrt{4})^5\). This means first taking the square root of 4 and then raising it to the fifth power. This conversion can make calculations more straightforward and connects back to the realm of radical expressions, showing how various exponent forms are related.

Radical Expressions

Radical expressions use roots, such as square roots, cube roots, and so on. They can often be viewed as expressions with fractional exponents, as they represent the same operations. Converting between radical forms and exponentiation forms is a useful skill:

• For instance, the square root of a number like \(\sqrt{4}\) is equivalent to 4 raised to the power of \(1/2\).
• This equivalence is due to the relationship of fractional exponents where the denominator as a root factor.

In this context, simplifying a radical expression often involves breaking down the base into more manageable parts. For the expression \((\sqrt{4})^5\), first calculate the square root of 4, which is 2, and then raise it to the power of 5 to get \(2^5 = 32\). Transforming radical expressions into their simpler forms using the power of exponents is essential for more advanced mathematics.