Exponents

Concept: This problem involves simplifying an expression with square roots and negative exponents using the laws of exponents. Key laws of exponents:


or


(for ) Step 1: Rewrite terms with negative exponents We are given the expression: Using :


So the expression becomes: Step 2: Combine the square roots Using the property , we can combine the terms under a single square root: Step 3: Simplify the expression inside the square root Multiply the fractions inside the square root: We can cancel out common terms in the numerator and the denominator:
The 'a' in the numerator cancels with the 'a' in the denominator.
The 'b' in the numerator cancels with the 'b' in the denominator.
The 'c' in the numerator cancels with the 'c' in the denominator. So, the expression inside the square root simplifies to: Step 4: Calculate the final value The expression becomes: And the square root of 1 is 1. Therefore, .

Concept: Prime factorization is the process of finding which prime numbers multiply together to make the original number. The exponent of a prime factor is the number of times that prime factor appears in the factorization. Step 1: Perform prime factorization of 864 Start by dividing 864 by the smallest prime numbers. Now 27 is not divisible by 2. Try the next prime number, 3. So, the prime factorization of 864 is . Step 2: Write the prime factorization in exponential form Step 3: Identify the exponent of 3 In the prime factorization , the prime factor 3 has an exponent of 3. Therefore, the exponent of 3 in the prime factorization of 864 is 3.