Accuplacer Radical and Absolute Value Equations — practice questions

1. A rational exponent is a fraction written where an exponent would go. 2. The numerator of that fraction gives the power applied to the base. 3. The denominator gives the root taken of the base, so a denominator of 2 means a square root. _Source: OpenStax College Algebra (CC BY 4.0), section 2.6 Other Types of Equations_

1. An absolute value measures a distance, so it can never come out negative. 2. The right side here is negative 9, which no distance can equal. 3. So the equation has no solution at all. 4. Had the right side been positive there would have been two solutions, and zero would have given one. _Source: OpenStax College Algebra (CC BY 4.0), section 2.6 Other Types of Equations_

1. The radical is already alone on the left, so square both sides. 2. Squaring gives x plus 6 equal to x squared. 3. Rearranging gives x squared minus x minus 6 equal to 0. 4. Factoring gives the brackets (x minus 3) and (x plus 2). 5. So the proposed solutions are x equal to 3 and x equal to negative 2. 6. Test x equal to 3: the square root of 9 is 3, which matches the right side. 7. Test x equal to negative 2: the square root of 4 is 2, but the right side is negative 2, so it fails. 8. Squaring both sides created that second root, so it is extraneous and only x equal to 3 survives. 9. Keeping both values skips the check that squaring makes necessary. 10. Rejecting both values ignores that x equal to 3 does satisfy the original equation. _Source: OpenStax College Algebra (CC BY 4.0), section 2.6 Other Types of Equations_