Problem 20 - Elementary and Intermediate Algebra

Question

Simplify each expression. $$ \sqrt[4]{32} $$

Step-by-Step Solution

Answer

\( 2 \cdot \sqrt[4]{2} \)

Step 1: Express the Number Under the Root

First, we express 32 as a power of 2, since it’s a factor of 32 and can be easily managed. We recognize that 32 can be written as, \[ 32 = 2^5 \].

Step 2: Rewrite the Expression Using Exponent Form

Replace the number inside the radical with its exponential form: \[ \sqrt[4]{32} = \sqrt[4]{2^5} \].

Step 3: Use the Property of Radicals

Apply the property of radicals, \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \), to convert the fourth root into an exponent: \[ (2^5)^{\frac{1}{4}} = 2^{\frac{5}{4}} \].

Step 4: Simplify the Fractional Exponent

Simplify the expression by separating integer powers from fractional powers, \[ 2^{\frac{5}{4}} = 2^{1 + \frac{1}{4}} = 2^1 \cdot 2^{\frac{1}{4}} \].

Step 5: Calculate the Integer Power

Calculate the integer component, which is straightforward: \[ 2^1 = 2 \].

Step 6: Express the Final Answer in Radical Form

Realize that the fractional part \(2^{\frac{1}{4}}\) is equivalent to the fourth root of 2. So, the expression becomes: \[ 2 \cdot \sqrt[4]{2} \].

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of the same number. When you see something like \( 2^5 \), it means that 2 is multiplied by itself 5 times: \( 2 \times 2 \times 2 \times 2 \times 2 = 32 \). This shorthand notation simplifies expressions and calculations. Exponents consist of two parts:

• The base (2 in this case), which is the number being multiplied.
• The exponent (5), which tells you how many times to multiply the base by itself.

Exponents allow us to easily manage and simplify large numbers or expressions involving multiplication. Remember that any number raised to the power of 0 is 1, which is a nifty property making calculations less complex.

Properties of Radicals

Radicals involve root functions, like square roots or cube roots. The radical symbol \( \sqrt{} \) often indicates these operations. An important property to remember is that radicals and exponents are closely connected. This is useful for simplifying expressions. A key property is \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \), which lets us convert radical expressions into fractions with exponents. This is handy because working with exponents is often more manageable than with radicals. Another crucial point is that:

• If two radicals have the same index (the little number outside the root symbol), you can multiply the radicands (the stuff inside the root) together under a single radical. For instance, \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
• If you have a radical raised to a power, the power can be applied either to the radicand first or to the entire expression, like \( (\sqrt{a})^n = (a^{\frac{1}{2}})^n \).

Understanding these properties lets you simplify expressions smoothly, making math just a bit less daunting.

Fractional Exponents

Fractional exponents are another way to represent roots. They can make complex root operations easier to deal with and appear when converting between radicals and exponents. Here, the numerator of the fraction is the power, and the denominator is the root. For example, \( a^{\frac{m}{n}} \) implies that you're finding the \( n \)-th root of \( a^m \). This could be thought of as \( \sqrt[n]{a^m} \), but written as an exponent.To understand the simplification:

• Convert \( n \)-th roots to fractional exponents using the formula \( \sqrt[n]{a} = a^{\frac{1}{n}} \).
• Multiply fractional exponents just like you would with whole numbers: \( a^{\frac{m}{n}} \cdot a^{\frac{p}{q}} = a^{\frac{mq + pn}{nq}} \).

This method helps with simplifying radicals, as seen with \( \sqrt[4]{32} = 2^{\frac{5}{4}} \), which can further simplify to a more digestible expression.