Which of the following is the simplified form of [tex]$\sqrt[7]{x} \cdot \sqrt[7]{x} \cdot \sqrt[7]{x}$[/tex]?
A. [tex]$x^{\frac{3}{7}}$[/tex]
B. [tex]$x^{\frac{1}{7}}$[/tex]
C. [tex]$x^{\frac{3}{21}}$[/tex]
D. [tex]$\sqrt[21]{x}$[/tex]
Answer (2)
Rewrite the expression using fractional exponents: 7 x = x 7 1 .
Apply the product of powers rule: x 7 1 ⋅ x 7 1 ⋅ x 7 1 = x 7 1 + 7 1 + 7 1 .
Simplify the exponent: x 7 1 + 7 1 + 7 1 = x 7 3 .
The simplified form of the expression is x 7 3 .
Explanation
Understanding the Problem We are given the expression 7 x \t ⋅ 7 x ⋅ 7 x and we need to simplify it.
Rewriting with Fractional Exponents Recall that n x can be written as x n 1 . Therefore, 7 x = x 7 1 .
Substituting Fractional Exponents The given expression can be rewritten as x 7 1 ⋅ x 7 1 ⋅ x 7 1 .
Applying the Rule of Exponents Using the rule of exponents, when multiplying terms with the same base, we add the exponents: x a ⋅ x b = x a + b .
Simplifying the Exponent Applying this rule, we have x 7 1 ⋅ x 7 1 ⋅ x 7 1 = x 7 1 + 7 1 + 7 1 = x 7 3 .
Final Answer Therefore, the simplified form of the given expression is x 7 3 .
Examples
Understanding fractional exponents and their simplification is crucial in various fields, such as physics and engineering, where you might encounter equations involving roots and powers. For instance, when calculating the period of a pendulum, you use the formula T = 2 π g L , where T is the period, L is the length, and g is the acceleration due to gravity. Simplifying such expressions using exponent rules helps in solving for unknown variables efficiently.
The simplified form of the expression 7 x ⋅ 7 x ⋅ 7 x is x 7 3 . Thus, the correct answer is option A: x 7 3 .
;
