PSAT Math Flashcards: Radicals And Absolute Values

What this deck covers

This deck focuses on Radicals And Absolute Values, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the solution set of $|x-2|<3$ written as a compound inequality?

Answer: $-1<x<5$. $-3 < x-2 < 3$ adds 2 to all parts: $-1 < x < 5$.

Flashcard 2: What is the quotient rule for radicals when $a\ge 0$ and $b>0$?

Answer: $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. Quotient under radical can be split into quotient of radicals.

Flashcard 3: What is the value of $\sqrt{(-5)^2}$?

Answer: $5$. $\sqrt{(-5)^2} = \sqrt{25} = 5 = |-5|$.

Flashcard 4: What is the value of $\frac{\sqrt{18}}{\sqrt{2}}$ in simplest form?

Answer: $3$. $\frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3$.

Flashcard 5: What is the value of $\sqrt{\frac{1}{49}}$?

Answer: $\frac{1}{7}$. Since $(\frac{1}{7})^2 = \frac{1}{49}$, this is the square root.

Flashcard 6: What is the simplified value of $\sqrt{12}\cdot\sqrt{3}$?

Answer: $6$. Using the product rule: $\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6$.

Flashcard 7: What is the value of $|{-17}|$?

Answer: $17$. Absolute value makes negative numbers positive.

Flashcard 8: What is the value of $| -17 |$?

Answer: $17$. Absolute value makes negative numbers positive.

Flashcard 9: What is the value of $|{-12}|$?

Answer: $12$. Absolute value removes the negative sign.

Flashcard 10: What is the value of $\sqrt{(-4)^2}$?

Answer: $4$. $(-4)^2 = 16$, so $\sqrt{16} = 4$.

Flashcard 11: What is the solution set of $|x|<5$?

Answer: $-5<x<5$. $|x| < 5$ means $x$ is less than 5 units from zero.

Flashcard 12: What is the key product identity for conjugates $(a+\sqrt{b})(a-\sqrt{b})$?

Answer: $(a+\sqrt{b})(a-\sqrt{b})=a^2-b$. Multiplying conjugates eliminates the radical terms.

Flashcard 13: What is the solution set of $|2x+1|=9$?

Answer: $x=4$ or $x=-5$. $2x+1 = 9$ gives $x=4$; $2x+1 = -9$ gives $x=-5$.

Flashcard 14: What is the solution set of $\sqrt{x-1}=3$ over the reals?

Answer: $x=10$. Square both sides: $x-1 = 9$, so $x = 10$.

Flashcard 15: What is the solution set of $|x-4|=3$?

Answer: $x=7$ or $x=1$. $x - 4 = 3$ or $x - 4 = -3$, giving $x = 7$ or $x = 1$.

Flashcard 16: What is the product $\sqrt{3}\cdot\sqrt{12}$ simplified?

Answer: $6$. $\sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6$.

Flashcard 17: What is the product rule for radicals: $\sqrt{a}\,\sqrt{b}$ for $a\ge 0$ and $b\ge 0$?

Answer: $\sqrt{a}\,\sqrt{b}=\sqrt{ab}$. Radicals can be multiplied under one radical sign.

Flashcard 18: Solve the inequality for $x$: $|x-2|\ge 3$.

Answer: $x\le -1$ or $x\ge 5$. $|x-2|\ge 3$ means $x-2\le -3$ or $x-2\ge 3$.

Flashcard 19: What is $\frac{1}{\sqrt{5}}$ with a rationalized denominator?

Answer: $\frac{\sqrt{5}}{5}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to get $\frac{\sqrt{5}}{5}$.

Flashcard 20: What is the simplified form of $\sqrt{12x^2}$ assuming $x \ge 0$?

Answer: $2x\sqrt{3}$. Factor out perfect squares: $\sqrt{12x^2} = \sqrt{4 \cdot 3 \cdot x^2} = 2x\sqrt{3}$.

Flashcard 21: What is the solution set of $|2x+1|=9$?

Answer: $x=4$ or $x=-5$. $2x + 1 = 9$ or $2x + 1 = -9$, solving gives $x = 4$ or $x = -5$.

Flashcard 22: What is the value of $|x-4|$ when $x=1$?

Answer: $3$. $1 - 4 = -3$, and $|-3| = 3$.

Flashcard 23: What is $|{-17}|$?

Answer: $17$. Absolute value removes the negative sign.

Flashcard 24: What is the simplified form of $\sqrt{ab}$ when $a\ge^0$ and $b\ge^0$?

Answer: $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$. Product rule: square root of a product equals product of square roots.

Flashcard 25: What is the solution set of $|x-3|=5$?

Answer: $x=8$ or $x=-2$. $x-3 = 5$ gives $x=8$; $x-3 = -5$ gives $x=-2$.

Flashcard 26: What is the value of $\sqrt{a^2}$ for a real number $a$?

Answer: $|a|$. The square root of a square gives the absolute value.

Flashcard 27: What is the solution set of $|x-2|\ge 4$ written with $\le$ or $\ge$?

Answer: $x\le -2$ or $x\ge 6$. $|x - 2| \ge 4$ means $x - 2 \le -4$ or $x - 2 \ge 4$.

Flashcard 28: What is the value of $\sqrt{0.01}$?

Answer: $0.1$. Since $0.1^2 = 0.01$, the square root of 0.01 is 0.1.

Flashcard 29: What is the value of $\sqrt{18}\cdot\sqrt{2}$?

Answer: $6$. Using the product rule: $\sqrt{18} \cdot \sqrt{2} = \sqrt{36} = 6$.

Flashcard 30: What is the definition of $|x|$ in terms of $x$ and $-x$?

Answer: $|x|$ is the distance from $0$; equals $x$ if $x \ge 0$, else $-x$. Absolute value removes the sign, making negatives positive.

Flashcard 31: What is the simplified form of $\sqrt{ab}$ for $a\ge^0$ and $b\ge^0$?

Answer: $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$. Product rule: square root of a product equals product of square roots.

Flashcard 32: What is the value of $\sqrt{\frac{1}{16}}$?

Answer: $\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, this is the square root.

Flashcard 33: What is the solution set of $|x|\le 3$ written as an interval?

Answer: $[-3,3]$. Distance from 0 at most 3 includes the endpoints.

Flashcard 34: What is the definition of absolute value $|x|$ written as a piecewise function?

Answer: $|x|=\begin{cases}x,&x\ge 0\\-x,&x<0\end{cases}$. Absolute value returns the distance from zero, always non-negative.

Flashcard 35: What is the definition of $|x|$ in terms of cases?

Answer: $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value keeps positive values unchanged, makes negative values positive.

Flashcard 36: What is the value of $\sqrt{50}$ in simplest radical form?

Answer: $5\sqrt{2}$. $50 = 25 \cdot 2 = 5^2 \cdot 2$, so $\sqrt{50} = 5\sqrt{2}$.

Flashcard 37: What is the value of $|{-12}|$?

Answer: $12$. Absolute value makes negative numbers positive.

Flashcard 38: What is the simplified form of $\frac{5}{\sqrt{3}}$ with no radical in the denominator?

Answer: $\frac{5\sqrt{3}}{3}$. Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ to rationalize: $\frac{5\sqrt{3}}{3}$.

Flashcard 39: What is the domain of $\sqrt{x}$ in the real numbers?

Answer: $x\ge^0$. Square roots of negative numbers are not real, so input must be non-negative.

Flashcard 40: What is the solution set of $|x|=7$?

Answer: $x=7$ or $x=-7$. The values that are 7 units from zero are $\pm 7$.

Flashcard 41: What is the solution set of $|x-2|\le^5$?

Answer: $-3\le x\le^7$. Values within 5 units of 2: $-5 \le x-2 \le 5$.

Flashcard 42: What is the definition of $\sqrt{a}$ for $a \ge 0$?

Answer: $\sqrt{a}$ is the nonnegative number whose square is $a$. The square root gives the principal (nonnegative) root only.

Flashcard 43: What is the value of $|\!-7|$?

Answer: $7$. Absolute value makes negative numbers positive: $|-7| = 7$.

Flashcard 44: What is the rationalized form of $\frac{3}{\sqrt{5}}$?

Answer: $\frac{3\sqrt{5}}{5}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to rationalize the denominator.

Flashcard 45: What is the value of $\sqrt{0.81}$?

Answer: $0.9$. Since $(0.9)^2 = 0.81$.

Flashcard 46: What is the simplified form of $\sqrt{72}$?

Answer: $6\sqrt{2}$. Factor out perfect squares: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

Flashcard 47: What is the simplified value of $\sqrt{a^2}$ for real $a$?

Answer: $\sqrt{a^2}=|a|$. Square root of a squared value gives the absolute value.

Flashcard 48: What is $\sqrt{x^2}$ when $x<0$?

Answer: $-x$. When $x < 0$, $|x| = -x$, and $\sqrt{x^2} = |x|$.

Flashcard 49: What is the value of $|{-x}|$ in simplest form?

Answer: $|{-x}|=|x|$. Absolute value measures distance from zero, regardless of sign.

Flashcard 50: What is the simplified form of $\sqrt{\frac{9}{16}}$?

Answer: $\frac{3}{4}$. Apply the quotient rule: $\sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4}$.

Flashcard 51: What is the value of $\sqrt{144}$?

Answer: $12$. Since $12^2 = 144$.

Flashcard 52: What is the simplified form of $\sqrt{50x^2}$ for real $x$?

Answer: $5|x|\sqrt{2}$. Factor: $50x^2 = 25 \cdot 2 \cdot x^2$, and $\sqrt{x^2} = |x|$.

Flashcard 53: What is the solution set of $|2x+1|=7$?

Answer: $x=3$ or $x=-4$. Solve $2x+1 = 7$ and $2x+1 = -7$.

Flashcard 54: What is the distance between $a$ and $b$ on a number line in terms of absolute value?

Answer: $|a-b|$. Distance is always positive, measured by absolute difference.

Flashcard 55: What is the simplified value of $|3-10|$?

Answer: $7$. $3-10 = -7$, and $|-7| = 7$.

Flashcard 56: What is the value of $\sqrt{12}\cdot\sqrt{3}$?

Answer: $6$. Apply product rule: $\sqrt{12 \cdot 3} = \sqrt{36} = 6$.

Flashcard 57: What is the value of $|x|^2$ in simplest form?

Answer: $|x|^2=x^2$. Squaring eliminates the sign, so $|x|^2$ and $x^2$ are equivalent.

Flashcard 58: What is the definition of $\sqrt{a}$ for $a \ge 0$?

Answer: $\sqrt{a}$ is the nonnegative number $r$ such that $r^2=a$. The square root gives the positive value whose square equals the radicand.

Flashcard 59: What is the product rule for radicals when $a\ge 0$ and $b\ge 0$?

Answer: $\sqrt{ab}=\sqrt{a}\,\sqrt{b}$. Product under radical can be split into product of radicals.

Flashcard 60: What is the solution set of $|x|=7$?

Answer: $x=7$ or $x=-7$. Absolute value equation has two solutions: positive and negative.

Flashcard 61: Identify the simplified form of $\sqrt{72}$.

Answer: $6\sqrt{2}$. $\sqrt{72}=\sqrt{36\cdot 2}=\sqrt{36}\cdot\sqrt{2}=6\sqrt{2}$.

Flashcard 62: What is the product rule for radicals when $a \ge 0$ and $b \ge 0$?

Answer: $\sqrt{ab}=\sqrt{a}\sqrt{b}$. The square root of a product equals the product of square roots.

Flashcard 63: What is the solution set of $|x|=5$?

Answer: $x=\pm^5$. Both $x = 5$ and $x = -5$ satisfy $|x| = 5$.

Flashcard 64: What is the solution set of $|x|<5$ written as a compound inequality?

Answer: $-5<x<5$. $|x| < 5$ means $x$ is between $-5$ and $5$.

Flashcard 65: What is the solution set of $|x|<3$?

Answer: $-3<x<3$. All values within 3 units of zero.

Flashcard 66: What is the simplified form of $\sqrt{50}$?

Answer: $5\sqrt{2}$. Factor out perfect squares: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

Flashcard 67: What is the solution set of $|x|=9$?

Answer: $x=9$ or $x=-9$. $|x| = 9$ means $x = 9$ or $x = -9$.

Flashcard 68: What is the solution set of $|2x+1|=7$?

Answer: $x=3$ or $x=-4$. $2x + 1 = 7$ or $2x + 1 = -7$, giving $x = 3$ or $x = -4$.

Flashcard 69: What is the simplified form of $\sqrt{18x^2}$ for $x \ge 0$?

Answer: $3x\sqrt{2}$. $\sqrt{18x^2} = \sqrt{9 \times 2 \times x^2} = 3|x|\sqrt{2} = 3x\sqrt{2}$ since $x \ge 0$.

Flashcard 70: What is the value of $\sqrt{x^2}$ for real $x$?

Answer: $|x|$. $\sqrt{x^2} = |x|$ because the result must be nonnegative.

Flashcard 71: What is the solution set of $|x-4|=3$?

Answer: $x=7$ or $x=1$. $x - 4 = 3$ or $x - 4 = -3$, giving $x = 7$ or $x = 1$.

Flashcard 72: What is the quotient $\frac{\sqrt{45}}{\sqrt{5}}$ simplified?

Answer: $3$. $\frac{\sqrt{45}}{\sqrt{5}} = \sqrt{\frac{45}{5}} = \sqrt{9} = 3$.

Flashcard 73: What is the simplified value of $\frac{\sqrt{48}}{\sqrt{3}}$?

Answer: $4$. Using the quotient rule: $\frac{\sqrt{48}}{\sqrt{3}} = \sqrt{\frac{48}{3}} = \sqrt{16} = 4$.

Flashcard 74: What is the definition of absolute value $|x|$?

Answer: $|x|=x$ if $x\ge 0$ and $|x|=-x$ if $x<0$. Absolute value gives the distance from zero on the number line.

Flashcard 75: What is the value of $\sqrt{72}$ in simplest radical form?

Answer: $6\sqrt{2}$. $72 = 36 \cdot 2 = 6^2 \cdot 2$, so $\sqrt{72} = 6\sqrt{2}$.

Flashcard 76: What is the simplified value of $\sqrt{(-5)^2}$?

Answer: $5$. $(-5)^2 = 25$, and $\sqrt{25} = 5$ (principal root is positive).

Flashcard 77: What is the simplified form of $\frac{\sqrt{45}}{\sqrt{5}}$?

Answer: $3$. Using the quotient rule: $\frac{\sqrt{45}}{\sqrt{5}} = \sqrt{\frac{45}{5}} = \sqrt{9} = 3$.

Flashcard 78: What is the value of $\sqrt{\frac{1}{16}}$?

Answer: $\frac{1}{4}$. Since $(\frac{1}{4})^2 = \frac{1}{16}$, the square root is $\frac{1}{4}$.

Flashcard 79: What is the value of $\sqrt{0}$?

Answer: $0$. Since $0^2 = 0$, the square root of zero is zero.

Flashcard 80: What is the solution set of $|x-2|=3$?

Answer: $x=-1\text{ or }x=5$. $x - 2 = 3$ gives $x = 5$; $x - 2 = -3$ gives $x = -1$.

Flashcard 81: What is the value of $\sqrt{72}$ in simplest radical form?

Answer: $6\sqrt{2}$. $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

Flashcard 82: What is the value of $\sqrt{16x^2}$ for real $x$?

Answer: $4|x|$. $\sqrt{16x^2} = \sqrt{16} \cdot \sqrt{x^2} = 4|x|$.

Flashcard 83: What is the solution set of $|2x-5|=7$?

Answer: $x=-1$ or $x=6$. Solve $2x-5 = 7$ and $2x-5 = -7$ to get both solutions.

Flashcard 84: What is the simplified value of $| -12 |$?

Answer: $12$. Absolute value removes the negative sign.

Flashcard 85: What is the principal value of $\sqrt{81}$?

Answer: $9$. Since $9^2 = 81$, the principal square root is $9$.

Flashcard 86: What is the value of $\sqrt{50}$ in simplest radical form?

Answer: $5\sqrt{2}$. $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

Flashcard 87: What is the solution set of $|x|\ge 3$ written using inequalities?

Answer: $x\le -3$ or $x\ge 3$. Values at least 3 units from zero are outside $[-3, 3]$.

Flashcard 88: What is the solution set of $|x-4|=3$?

Answer: $x=7$ or $x=1$. $x-4 = 3$ or $x-4 = -3$, giving $x = 7$ or $x = 1$.

Flashcard 89: What is the value of $|-7|$?

Answer: $7$. Absolute value makes negative numbers positive.

Flashcard 90: What is the definition of $\sqrt{x}$ for real numbers $x$?

Answer: $\sqrt{x}$ is the nonnegative real number $r$ with $r^2=x$. The principal square root is always nonnegative by definition.

Flashcard 91: What is the value of $\sqrt{18x^2}$ in simplest form for all real $x$?

Answer: $3|x|\sqrt{2}$. $\sqrt{x^2} = |x|$, so $\sqrt{18x^2} = 3|x|\sqrt{2}$.

Flashcard 92: What is the solution set of $|x|=5$?

Answer: $x=5$ or $x=-5$. The equation $|x| = 5$ means $x$ is 5 units from zero.

Flashcard 93: What is the solution set of $|x|<4$ written as an interval?

Answer: $(-4,4)$. $|x| < 4$ means $-4 < x < 4$.

Flashcard 94: What is the simplified form of $\sqrt{\frac{48}{3}}$?

Answer: $4$. $\sqrt{\frac{48}{3}} = \sqrt{16} = 4$.

Flashcard 95: What is the definition of $|x|$ in terms of a piecewise rule?

Answer: $|x|=x$ if $x\ge 0$; $|x|=-x$ if $x<0$. Absolute value keeps positive values unchanged, makes negatives positive.

Flashcard 96: What is the value of $\sqrt{49}$?

Answer: $7$. Since $7^2 = 49$, the square root of 49 is 7.

Flashcard 97: What is the simplified value of $\sqrt{72}$?

Answer: $6\sqrt{2}$. $72 = 36 \times 2 = 6^2 \times 2$, so $\sqrt{72} = 6\sqrt{2}$.

Flashcard 98: What is the solution set of $|x-3|=4$?

Answer: $x=7$ or $x=-1$. Solve $x-3 = 4$ and $x-3 = -4$.

Flashcard 99: What is the simplified value of $\sqrt{\frac{49}{64}}$?

Answer: $\frac{7}{8}$. $\sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8}$.

Flashcard 100: What is the rule for a negative exponent with radicals: $a^{-\frac{m}{n}}$?

Answer: $a^{-\frac{m}{n}}=\frac{1}{\sqrt[n]{a^m}}$. Negative exponent means reciprocal of the positive exponent form.