Problem 66 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ \frac{1}{2} \sqrt{32} $$

Step-by-Step Solution

Answer

The simplified form of \( \frac{1}{2} \sqrt{32} \) is \( 2 \sqrt{2} \).

Step 1: Simplify the Radicand

First break down \( 32 \) inside the square root into its prime factors. Here, \( 32 = 2^5 \). So substitute \( 32 \) inside the root with \( 2^5 \). We then have \( \frac{1}{2} \sqrt{2^5} \).

Step 2: Apply the Rules of Square Roots

Remember that \( \sqrt{a^n} = a^{n/2} \) if a > 0 and n is even. Applying this, \( \sqrt{2^5} \) simplifies to \( 2^{5/2} \). Now, simplify to have \( \frac{1}{2} \cdot 2^{5/2} \).

Step 3: Simplify the Fraction and the Power

Rewrite \( 2^{5/2} \) into \( 2^2 \cdot 2^{1/2} \), we get \( 2^{2} \cdot 2^{1/2} = 4 \cdot \sqrt{2} \). Then, distribute the fraction over the multiplication, which results in \( 2 \cdot \sqrt{2} \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics and are used to find a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. The symbol used for square roots is '√', which is called the radical sign.

If you have an expression like \( \sqrt{32} \), you want to simplify it by finding the largest square factor. In this case, it's beneficial to use prime factorization. This simplification turns complex square roots into simpler expressions with real numbers or simpler radicals, making calculations easier and results clearer.

Understanding the properties of square roots can help simplify various STEM problems and aids in performing mental math quickly.

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime number components. Prime numbers are numbers that have only two distinct positive divisors: 1 and themselves, such as 2, 3, 5, 7, etc. This method is instrumental in simplifying radical expressions and finding the greatest common divisor or least common multiple of numbers.

To factor a number into primes, divide it by the smallest prime number until the quotient is 1. For example, with the number 32, divide by the smallest prime, 2, repetitively: 32 ÷ 2 = 16, 16 ÷ 2 = 8, 8 ÷ 2 = 4, 4 ÷ 2 = 2. Thus, 32 can be expressed as \( 2^5 \).

This breakdown makes simplifying radicals straightforward, as you can then apply root rules to these prime components.

Exponents and Powers

Exponents and powers are a mathematical shorthand used for expressing numbers multiplied by themselves a certain number of times. In the expression \( a^n \), 'a' is the base, and 'n' is the exponent. This means 'a' is multiplied by itself 'n' number of times.

In simplifying radical expressions, powers become crucial. For instance, when simplifying \( \sqrt{2^5} \), understanding that \( \sqrt{a^n} = a^{n/2} \) assists in breaking down the expression further. It turns into \( 2^{5/2} \), which can be rewritten using properties of exponents.

Knowing how to manipulate expressions with exponents is key to solving complex algebraic problems efficiently and understanding the underlying mathematics better.