Problem 16 - Elementary Algebra

Question

For the following problems, simplify each expressions. $$ \frac{\sqrt{96}}{\sqrt{24}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\frac{\sqrt{96}}{\sqrt{24}}$. Answer: The simplified expression is 2.

Step 1: Simplify roots

Firstly, let us rewrite the numbers under the square root as products of their prime factors: $$ \frac{\sqrt{96}}{\sqrt{24}} = \frac{\sqrt{2^5 \cdot 3^1}}{\sqrt{2^3 \cdot 3^1}} $$

Step 2: Divide the roots

Now, by applying the rules of radicals, we can simplify each square root and divide: $$ \frac{\sqrt{2^5 \cdot 3^1}}{\sqrt{2^3 \cdot 3^1}} = \frac{2^{\frac{5}{2}} \cdot 3^{\frac{1}{2}}}{2^{\frac{3}{2}} \cdot 3^{\frac{1}{2}}} $$

Step 3: Cancel common factors

We can now cancel the common factors in the numerator and denominator: $$ \frac{2^{\frac{5}{2}} \cdot 3^{\frac{1}{2}}}{2^{\frac{3}{2}} \cdot 3^{\frac{1}{2}}} = \frac{2^{\frac{5}{2} - \frac{3}{2}} \cdot 3^{\frac{1}{2} - \frac{1}{2}}}{1} $$

Step 4: Simplify exponential terms

Finally, we can simplify the exponential terms and obtain the simplified expression: $$ \frac{2^{\frac{5}{2} - \frac{3}{2}} \cdot 3^{\frac{1}{2} - \frac{1}{2}}}{1} = \frac{2^{\frac{2}{2}} \cdot 3^0 }{1} = \frac{2^1 \cdot 1}{1} = 2 $$ So, the simplified expression is 2.

Key Concepts

Radicals Simplification

Simplifying radicals, or roots, is a fundamental part of algebra that deals with reducing expressions to their simplest form. A square root, denoted by the radical symbol \( \sqrt{} \), represents a number which, when multiplied by itself, yields the original number under the radical. For instance, \( \sqrt{25} \) equals 5 because 5 times 5 is 25.

To simplify square roots like \( \sqrt{96} \) or \( \sqrt{24} \) in the given exercise, we use prime factorization to break down the numbers into products of their prime factors and then apply properties of exponents. In this case, recognising that each pair of same prime factors can be taken out of the radical as a single factor is crucial. When we have a fraction, the process involves simplifying both the numerator and the denominator separately, and then, if possible, reducing the fraction to its simplest form.

Prime Factorization

Prime factorization is a technique that expresses a number as the product of its prime factors. Primes are numbers greater than 1 that have no divisors other than 1 and themselves. By breaking a composite number into the product of prime numbers, we can simplify mathematical expressions, especially when dealing with exponents and roots.

To perform prime factorization, divide the original number by the smallest prime number possible, usually starting with 2, and continue the process with the quotient until all factors are prime. For example, the prime factorization of 96 is \( 2^5 \cdot 3^1 \) and for 24 it is \( 2^3 \cdot 3^1 \). While prime factorization is particularly helpful for simplifying square roots, it is also used in other areas such as finding the greatest common divisor, least common multiple, or simplifying fractions.

Exponents and Powers

Exponents and powers are expressions that represent repeated multiplication. The exponent, situated above and to the right of the base in a superscript position, indicates the number of times the base is multiplied by itself. For instance, \( 2^3 \) (read as 'two cubed' or 'two to the third power') equals 8, because 2 is multiplied by itself three times (2 \( \cdot \) 2 \( \cdot \) 2).

When simplifying square roots with exponents, it's essential to use the properties of exponents. For example, \( 2^{\frac{5}{2}} \) becomes 2 raised to the power of 5 divided by 2, which simplifies to 2 squared. Also, any number raised to the power of 0 is 1, indicated by \( 3^0 = 1 \). Hence, understanding how to manipulate exponents effectively unlocks the ability to simplify complex radical expressions and solve algebraic problems accurately.