Correct Answer
Options B (x to the sixth power y to the negative second power z to the one half power), \( (x^6y^{-2}z^{\frac{1}{2}}), \) D (fraction with numerator of x to the sixth power z to the one-half power and denominator of y squared), \( (\frac {x^6z^{\frac{1}{2}}} {y^2}), \) and E (x to the sixth power y to the negative second power the square root of z) \( (x^6y^{-2}\sqrt{z}) \) should be selected.
Explanation
Exponent rules, which are also known as the "laws of exponents" or the "properties of exponents," make simplifying expressions with exponents easier. The rules can help you simplify expressions like the one in this problem that have fractions and negative numbers as their exponents.
For example, the negative exponent rule says that a variable like x raised to the power of a negative exponent in the denominator is equal to that variable in the numerator raised to the opposite of that power. For example, fraction with numerator of one and denominator of x raised to the negative fourth power equals x raised to the fourth power. \( \frac{1}{x^{-4}}=x^4. \) This helps here because you can move the x raised to the negative fourth power \( x^{-4} \) in the denominator to the numerator of the fraction as x raised to the fourth power. \( x^4. \)
Now you have an x squared \( x^2 \) and an x raised to the fourth power \( x^4 \) in the numerator. The product rule says that any variable raised to a power times that variable raised to another power equals that variable raised to the sum of the two powers. In math terms, x raised to the power of m times x raised to the power of n equals x raised to the power of m plus n. \( x^m+x^n=x^{m+n}. \) Since two plus four equals six \( 2+4=6 \) and there are no more x terms to consider, any equivalent expression must have x raised to the power of 6 \( x^6 \) in the numerator. That eliminates options C and F.
Apply this same rule to y and z. The variable y is raised to the third power in the numerator and fifth power in the denominator. Simplify the y term by subtracting the 3 in the numerator from the 5 in the denominator, leaving y squared \( y^2 \) in the denominator (or y raised to the power of negative 2 \( y^{-2} \) in the numerator). And simplify the z term by subtracting the 2 from the z squared \( z^2 \) in the denominator from five halves, \( \frac{5}{2}, \) the exponent on the z term in the numerator, leaving z raised to the power of one half \( z^{\frac{1}{2}} \) in the numerator (option D).
Finally, the fractional exponent rule says that any variable raised to the one half \( \frac{1}{2} \) power is equal to the square root of that variable. Thus, z raised to the power of one half \( z^{\frac{1}{2}} \) equals the square root of z \( \sqrt{z} \) (option E).