Radicals & Absolute Values - SAT Math

1. Simplify $\frac{\text{√}18}{\text{√}2}$.

   $\text{√}9 = 3$. Simplify by dividing: $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9}$.

2. Simplify $\text{√}72$.

   $6\text{√}2$. Factor out perfect squares: $\sqrt{72} = \sqrt{36 \times 2}$.

3. What is the principal square root of 81?

   9. Since $9 \times 9 = 81$.

4. What is the absolute value of 0?

   0. Zero has no distance from itself.

5. What is the absolute value of -15?

   15. Absolute value makes any number non-negative.

6. What is the square root of 49?

   7. Since $7 \times 7 = 49$.

7. Solve for $x$: $|\text{x} + 5| = 8$.

   $x = 3$ or $x = -13$. Distance from -5 equals 8, so $x + 5 = \pm 8$.

8. Simplify: $\text{sqrt}(72)$.

   $6\text{sqrt}(2)$. Factor out perfect squares: $\sqrt{36 \times 2} = 6\sqrt{2}$.

9. State the property: $|\text{a}| \times |\text{b}| = ?$

   $|\text{a} \times \text{b}|$. Absolute values multiply to give the absolute value of the product.

10. Simplify $\frac{\text{√}45}{\text{√}5}$.

    $\text{√}9 = 3$. Divide radicals: $\frac{\sqrt{45}}{\sqrt{5}} = \sqrt{\frac{45}{5}} = \sqrt{9}$.

11. Solve for $x$: $|\text{3x} + 2| = 11$.

    $x = 3$ or $x = -\frac{13}{3}$. Distance equals 11, so $3x + 2 = \pm 11$.

12. Simplify $\text{√}50$.

    $5\text{√}2$. Factor out perfect squares: $\sqrt{50} = \sqrt{25 \times 2}$.

13. State the property: $|\text{a} \times \text{b}|$.

    $|\text{a} \times \text{b}| = |\text{a}| \times |\text{b}|$. The absolute value of a product equals the product of absolute values.

14. What is the absolute value of $|\text{3} - 7|$?

    4. Calculate $3 - 7 = -4$, then take absolute value.

15. What is $\text{√}64$?

    8. Since $8 \times 8 = 64$.

16. Solve for $x$: $\text{√}x = 5$.

    $x = 25$. Square both sides to eliminate the radical.

17. Simplify: $\sqrt{48} + \sqrt{12}$.

    $6\sqrt{3}$. Simplify to $4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$.

18. What is the absolute value of $|\text{-7}|$?

    7. Absolute value makes negative numbers positive.

19. Simplify $\frac{\sqrt{18}}{\sqrt{2}}$.

    $\sqrt{9} = 3$. Simplify by dividing: $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9}$.

20. What is the square root of 64?

    8. Since $8^2 = 64$, the principal square root is 8.

21. Solve for $x$: $|\text{x}| = 12$.

    $x = 12$ or $x = -12$. Absolute value equation has two solutions: $x = \pm 12$.

22. Simplify $\text{√}0.25$.

    0.5. Since $0.5 \times 0.5 = 0.25$.

23. What is the value of $|\text{-7}|$?

    7. Absolute value gives the distance from zero, always positive.

24. What is the result of $\text{√}1$?

    1. Since $1 \times 1 = 1$.

25. What is the value of $|-7|$?

    7. Absolute value gives the distance from zero, always positive.

26. What is the result of $|5 - 9|$?

    4. Calculate $5 - 9 = -4$, then take absolute value to get 4.

27. Simplify $\text{√}12 - \text{√}3$.

    $3\text{√}3 - \text{√}3 = 2\text{√}3$. Simplify radicals first: $\sqrt{12} = 2\sqrt{3}$.

28. What is $\text{√}(-9)$?

    Not a real number. Square roots of negative numbers aren't real.

29. What is the cube root of 27?

    3. Since $3^3 = 27$, the cube root is 3.

30. Express $\text{√}32$ in simplest radical form.

    $4\text{√}2$. Factor out perfect squares: $\sqrt{32} = \sqrt{16 \times 2}$.

31. Simplify $\text{√}20$.

    $2\text{√}5$. Factor out perfect squares: $\sqrt{20} = \sqrt{4 \times 5}$.

32. Solve for $x$: $|\text{x} - 4| = 3$.

    $x = 7$ or $x = 1$. Distance from 4 equals 3, so $x - 4 = \pm 3$.

33. Solve for $x$: $|x - 3| = 5$.

    $x = 8$ or $x = -2$. Distance from 3 equals 5, so $x - 3 = 5$ or $x - 3 = -5$.

34. Solve for $x$: $|\text{2x} - 3| = 5$.

    $x = 4$ or $x = -1$. Distance equals 5, so $2x - 3 = \pm 5$.

35. State the property: $|\text{a}| + |\text{b}| \neq ?$

    $|\text{a} + \text{b}|$. Triangle inequality shows they're not always equal.

36. Simplify $\text{√}8 + \text{√}18$.

    $2\text{√}2 + 3\text{√}2 = 5\text{√}2$. Simplify each radical first: $\sqrt{8} = 2\sqrt{2}$, $\sqrt{18} = 3\sqrt{2}$.

37. Simplify $\text{√}27$.

    $3\text{√}3$. Factor out perfect squares: $\sqrt{27} = \sqrt{9 \times 3}$.

38. Simplify: $\frac{\sqrt{50}}{\sqrt{2}}$.

    $\sqrt{25} = 5$. This simplifies to $\sqrt{25} = 5$ using the quotient property of radicals.

39. Simplify: $\sqrt{48} + \sqrt{12}$

    $6\sqrt{3}$. Simplify to $4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$

40. What is the result of $|\text{-25}|$?

    25. Absolute value removes the negative sign.

41. What is the square of $\text{√}3$?

    3. The square of a square root equals the radicand.

42. Simplify $\text{√}200$.

    $10\text{√}2$. Factor out perfect squares: $\sqrt{200} = \sqrt{100 \times 2}$.

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