Identify which of the following is not equivalent to [tex]a^{\frac{1}{4}}[/tex].
A) [tex]a^{\frac{1}{8}} \times a^{\frac{1}{8}}[/tex]
B) [tex]\sqrt{a}[/tex]
C) [tex]\left(a^{\frac{1}{8}}\right)^2[/tex]
D) [tex]a^{\frac{3}{4}} \div a^{\frac{1}{2}}[/tex]
Answer (1)
Simplify each expression using exponent rules.
Option A simplifies to a 4 1 .
Option B simplifies to a 2 1 .
Options C and D simplify to a 4 1 .
The expression not equivalent to a 4 1 is B .
Explanation
Understanding the Problem We are given the expression a 4 1 and four other expressions. Our goal is to identify which of the four expressions is not equivalent to a 4 1 .
Plan of Action Let's simplify each of the four expressions using exponent rules.
Simplifying Option A A) a 8 1 × a 8 1 = a 8 1 + 8 1 = a 8 2 = a 4 1
Simplifying Option B B) a = a 2 1
Simplifying Option C C) ( a 8 1 ) 2 = a 8 2 = a 4 1
Simplifying Option D D) a 4 3 ÷ a 2 1 = a 4 3 − 2 1 = a 4 3 − 4 2 = a 4 1
Identifying the Non-Equivalent Expression Comparing the simplified expressions to a 4 1 , we see that option B, a = a 2 1 , is not equivalent to a 4 1 .
Final Answer Therefore, the expression that is not equivalent to a 4 1 is a .
Examples
Understanding exponents and roots is crucial in many scientific fields. For instance, in physics, the kinetic energy of an object is related to the square of its velocity ( K E = 2 1 m v 2 ), and understanding how to manipulate exponents helps in solving for velocity when kinetic energy is known ( v = m 2 K E ). Similarly, in finance, compound interest calculations involve exponents, where the future value of an investment is calculated using the formula F V = P V ( 1 + r ) n , where r is the interest rate and n is the number of compounding periods. Mastering these concepts provides a solid foundation for advanced problem-solving in various disciplines.
